Presbyopia correction using patient data

ABSTRACT

Methods, devices, and systems establish an optical surface shape that mitigates or treats presbyopia in a particular patient. The combination of distance vision and near vision in a patient can be improved, often based on input patient parameters such as pupil size, residual accommodation, and power need. Iterative optimization may generate a customized corrective optical shape for the patient.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.11/938,648 filed Nov. 12, 2007, which is a continuation of U.S. patentapplication Ser. No. 10/738,358 filed Dec. 5, 2003, now U.S. Pat. No.7,293,873, which claims the benefit of U.S. Patent Application Ser. Nos.60/519,885 filed Nov. 13, 2003; 60/468,387 filed May 5, 2003; 60/468,303filed May 5, 2003; and 60/431,634 filed Dec. 6, 2002, the entiredisclosures of which are incorporated herein by reference for allpurposes.

BACKGROUND OF THE INVENTION

This invention generally relates to optical correction, and inparticular provides methods, devices, and systems for mitigating ortreating presbyopia and other vision conditions, often by scaling,deriving, or generating a prescription to treat a particular patient.

Presbyopia is a condition that affects the accommodation properties ofthe eye. As objects move closer to a young, properly functioning eye,the effects of ciliary muscle contraction and zonular relaxation allowthe lens of the eye to become rounder or more convex, and thus increaseits optical power and ability to focus at near distances. Accommodationcan allow the eye to focus and refocus between near and far objects.

Presbyopia normally develops as a person ages, and is associated with anatural progressive loss of accommodation, sometimes referred to as “oldsight.” The presbyopic eye often loses the ability to rapidly and easilyrefocus on objects at varying distances. There may also be a loss in theability to focus on objects at near distances. Although the conditionprogresses over the lifetime of an individual, the effects of presbyopiausually become noticeable after the age of 45 years. By the age of 65years, the crystalline lens has often lost almost all elastic propertiesand has only limited ability to change shape. Residual accommodationrefers to the amount of accommodation that remains in the eye. A lowerdegree of residual accommodation contributes to more severe presbyopia,whereas a higher amount of residual accommodation correlates with lesssevere presbyopia.

Known methods and devices for treating presbyopia seek to provide visionapproaching that of an emmetropic eye. In an emmetropic eye, bothdistant objects and near objects can be seen due to the accommodationproperties of the eye. To address the vision problems associated withpresbyopia, reading glasses have traditionally been used by individualsto add plus power diopter to the eye, thus allowing the eye to focus onnear objects and maintain a clear image. This approach is similar tothat of treating hyperopia, or farsightedness.

Presbyopia has also been treated with bi-focal eyeglasses, where oneportion of the lens is corrected for distance vision, and anotherportion of the lens is corrected for near vision. When peering downthrough the bifocals, the individual looks through the portion of thelens corrected for near vision. When viewing distant objects, theindividual looks higher, through the portion of the bi-focals correctedfor distance vision. Thus with little or no accommodation, theindividual can see both far and near objects.

Contact lenses and intra-ocular lenses (IOLs) have also been used totreat presbyopia. One approach is to provide the individual withmonovision, where one eye (usually the primary eye) is corrected fordistance-vision, while the other eye is corrected for near-vision.Unfortunately, with monovision the individual may not clearly seeobjects that are intermediately positioned because the object isout-of-focus for both eyes. Also, an individual may have trouble seeingwith only one eye, or may be unable to tolerate an imbalance betweentheir eyes. In addition to monovision, other approaches includebilateral correction with either bi-focal or multi-focal lenses. In thecase of bi-focal lenses, the lens is made so that both a distant pointand a near point can be focused. In the multi-focal case, there existmany focal points between near targets and far targets.

Surgical treatments have also been proposed for presbyopia. Anteriorsclerostomy involves a surgical incision into the sclera that enlargesthe ciliary space and facilitates movement of the lens. Also, scleralexpansion bands (SEBs) have been suggested for increasing the ciliaryspace. Problems remain with such techniques, however, such asinconsistent and unpredictable outcomes.

In the field of refractive surgery, certain ablation profiles have beensuggested to treat the condition, often with the goal of increasing therange of focus of the eye, as opposed to restoring accommodation in thepatient's eye. Many of these ablation profiles can provide a singleexcellent focus of the eye, yet they do not provide an increased depthof focus such that optimal distance acuity, optimal near acuity, andacceptable intermediate acuity occur simultaneously. Shapes have beenproposed for providing enhanced distance and near vision, yet currentapproaches do not provide ideal results for all patients.

In light of the above, it would be desirable to have improved methods,devices, and systems for treatment and/or mitigation of presbyopia andother optical defects. Optionally, it may be desirable to provideimproved prescriptions in the form of practical customized or optimizedprescription shapes for treating or mitigating vision conditions such aspresbyopia in a particular patient.

BRIEF SUMMARY OF THE INVENTION

The present invention provides improved devices, systems, and methodsfor mitigating or treating presbyopia and other vision conditions. Thepresent invention can establish a prescription that mitigates or treatspresbyopia in a particular patient. In some embodiments, an opticallyoptimized shape may be generated based on patient data input. Typically,the shape will represent a compromise between improved near vision andimproved distance vision. These optimized shapes can be derivednumerically using input patient parameters such as pupil size, residualaccommodation, and desired vergence. Presbyopia-mitigating shapes may bescaled (or otherwise varied) in response to patient data such as one ormore pupil diameters. Appropriate scaling may be determined at least inpart from prior patient data from patients having differing pupil sizesand/or differing shapes. Advantageously, presbyopia-mitigatingprescriptions may be derived from, scaled using, and/or optimized toprovide at least one desired optical power (and/or manifest power),often to provide a plurality of optical powers at differing viewingconditions, thereby taking advantage of changes in pupil size whenviewing objects under differing viewing conditions such as at differingdistances and lighting conditions.

In a first aspect, the invention provides a method for treating existingor potential presbyopia of a patient. The patient has an eye with apupil, a change in viewing distance with the eye inducing a change inpupil dimension. The method comprises measuring a first dimension of thepupil at a first viewing distance, and determining a first desired powerfor the eye at the first viewing distance. A prescription for the eye isdetermined such that the prescription provides the first desired powerwhen the pupil has the first dimension, and such that the prescriptioneffects a desired change in power in response to the change in pupildimension, the desired change in power mitigating the presbyopia.

In many embodiments, a rate of the desired change in power for thechange in pupil dimension comprises from about 0.25 D/mm to about 5.0D/mm. When the patient is about 45 years old or less, and the rate maycomprise from about 0.25 D/mm to about 1.0 D/mm. When the patient isabout 60 years old or less the rate may comprise from about 1.0 D/mm toabout 5.0 D/mm. A second desired optical power for the eye may bedetermined at a second viewing distance. At least a third desiredoptical power for the eye may also be determined, each optical powerhaving an associated viewing condition, with a rate of an incrementaldesired change in power for an incremental change in pupil size varyingwithin a pupil size range of the patient. The change in pupil dimensionof the patient may be measured by measuring a second pupil dimension ofthe pupil at the second viewing distance, and/or the rate of the desiredchange in optical power for the change in pupil dimension may be assumedto be consistent for a plurality of patients.

The eye may have a residual accommodation range, and the first desiredpower for the eye may be determined so that the eye adjusts within theresidual accommodation range when viewing at the first viewing distancewith the first desired optical power. Optionally, particularly when thepatient is about 60 years old or less, the first desired power for theeye and/or the desired change in power may be adjusted in response to ananticipated shrinkage of the pupil with age and/or anticipated reductionof residual accommodation.

The prescription may be determined at least in part by iterativelyoptimizing a goal function, by scaling a refractive shape, and/or byanalytically or numerically deriving an optical shape providing aplurality of desired optical powers at an associated plurality ofviewing conditions.

In a system aspect, the invention provides a system for treatingexisting or potential presbyopia of a patient. The patient has an eyewith a pupil, a change in viewing distance with the eye inducing achange in pupil dimension. The system comprises a pupilometer formeasuring a first dimension of the pupil while the eye is viewing at afirst viewing distance. A prescription generating module has an inputaccepting a desired power for the eye and the first dimension. Themodule determines a prescription for the eye providing a first desiredpower when the pupil has the first dimension, the prescription effectinga desired change in power in response to the change in pupil dimension.The desired change in power mitigates the presbyopia.

The prescription generating module may comprise an optimizer module thatdetermines the prescription based on the pupil diameter and the desiredpower using a goal function appropriate for the presbyopia; a scalingmodule that scales a central portion of a prescription shape based onthe pupil dimension such that the prescription shape amelioratespresbyopia, and such that the central portion has a dimension betweenabout 0.35 and about 0.55 of the pupil dimension; and/or a prescriptioncalculating module calculating a presbyopia-mitigating prescription forthe eye in response to the pupil dimension and the change in pupildimension so that the eye has the first desired power suitable for thefirst viewing distance and so that the eye has a second desired powerfor a second viewing distance. Optionally, a laser may impose theprescription on the eye, typically by ablating corneal tissue.

In another aspect, the invention provides a method for determining aprescription that mitigates or treats presbyopia in a particularpatient. The method comprises selecting a goal function appropriate forpresbyopia of an eye, inputting a set of patient parameters specific forthe particular patient, and determining an optical shape for theparticular patient appropriate for differing viewing conditions based onthe set of patient parameters per the goal function so as to mitigate ortreat the presbyopia in the patient.

The goal function can reflect optical quality throughout a vergencerange. The goal function may also comprise a ratio of an opticalparameter of the eye with a diffraction theory parameter. Relatedly, thegoal function may also comprise at least one parameter selected from thegroup consisting of Strehl Ratio (SR), modulation transfer function(MTF), point spread function (PSF), encircled energy (EE), MTF volume orvolume under MTF surface (MTFV), compound modulation transfer function(CMTF), and contrast sensitivity (CS). The goal function may also bebased on geometrical optics. Similarly, the goal function can bedetermined using ray tracing. In this context, the phrase ‘ray tracing’has a meaning identical to ‘geometrical optics’. The set of patientparameters can include at least one parameter selected from the groupconsisting of pupil size, residual accommodation, power need, andvergence. In this context the phrase “power need” has a meaningidentical to “vergence.”

The prescription may comprise an optical shape determined by inputting aset of patient parameters specific for the particular patient into anoptimizer. The shape is derived for the particular patient per a goalfunction so as to mitigate or treat the presbyopia in the patient. Aninitial optical shape can be input, the initial shape often beingradially symmetric. Relatedly, the radially symmetric shape may bedecomposed into a set of polynomials having at least two independentvariables. Further, one of the at least two independent variables can bethe ratio of the customized shape diameter to pupil diameter. Theiterative optimization may be selected from the group consisting ofDownhill Simplex method, Direction set method, and Simulated Annealingmethod, or the like. The set of patient parameters can include at leastone parameter selected from the group consisting of pupil size, residualaccommodation, and power need.

Optionally, the presbyopia may be treated by administering to thepatient a procedure selected from the group consisting of ablating acornea of the patient to provide a corneal shape that corresponds to theoptical shape, providing the patient with a contact lens or spectaclelens that has a shape that corresponds to the optical shape, andproviding the patient with an intra-ocular lens that has a shape thatcorresponds to the optical shape. The optical shape may be determinedbased at least in part on an expansion such as a regular polynomial(Even-Power-Term polynomials (“EPTP”) or non-EPTP), a Zernikepolynomial, a Fourier series, and a discrete shape entirety. Theexpansion may be a 3rd order or 4th order non-EPTP expansion, or a 6thor 8th order EPTP expansion. The optical shape may be determined basedat least in part on a presbyopia-add to pupil ratio (PAR), the PARranging from about 0.2 to about 1.0.

In another system aspect, the present invention provides a system forestablishing a prescription that mitigates or treats presbyopia in aparticular patient, where the system includes an input that accepts aset of patient parameters, and a module that determines an optical shapefor the particular patient based on the set of patient parameters, usinga goal function appropriate for presbyopia of an eye.

The module may include data processing software and/or hardware, and maybe optionally integrated with other data processing structures. Themodule may comprise an optimizer module that determines the prescriptionfor the particular patient based on the set of patient parameters, usinga goal function appropriate for presbyopia of an eye. A processor maygenerate an ablation profile, and a laser system can direct laser energyonto the cornea according to the ablation profile so as to reprofile asurface of the cornea from the first shape to the second shape, thesecond shape corresponding to the determined optical shape. Pupildiameters may be measured for input under one or more of the followingconditions: when focusing on a near object; when focusing on a distantobject; under photopic conditions; under mesopic conditions; underscotopic conditions. The prescription shape may be aspherical when thecentral portion of the prescription shape is aspherical; theprescription shape may be spherical when the central portion of theprescription shape is spherical; the prescription shape may beaspherical when the central portion of the prescription shape isspherical; and/or the prescription shape may be spherical when thecentral portion of the prescription shape is aspherical, with healingand LASIK flap effects and the like optionally varying the final shapeof the eye. The dimension of the prescription shape central portion maycomprise a diameter of the central portion and may remain within a rangebetween about 0.4 and about 0.5 of the pupil diameter of the particularpatient, or within a range between about 0.43 and about 0.46 of thepupil diameter of the particular patient; a power of the central portionis optionally between about 1.5 diopters and about 4.0 diopters (ideallybeing about 3.1 diopters).

In another aspect, the invention provides a method for treatingpresbyopia of an eye of a patient. The method comprises identifying afirst pupil size of the eye under a first viewing condition. A secondpupil size of the eye is identified under a second viewing condition. Apresbyopia-mitigating prescription is calculated for the eye in responseto the pupil sizes so that the eye has a first power suitable for thefirst viewing condition at the first size and so that the eye has asecond power suitable for the second viewing condition at the secondsize.

Calculating the prescription may comprise determining a first effectivepower of the eye with the first pupil size and calculating a secondeffective power of the eye with the second pupil size. The first andsecond pupil diameters may be measured from the eye of the patient whilethe eye is viewing with the first and second viewing conditions,respectively. The prescription often comprises a prescription shape, andthe method may include altering the refraction of the eye according tothe prescription shape. The refraction of the eye can be altered usingat least one of a laser, a contact lens, an intraocular lens, and aspectacle. One or more additional pupil diameters of the eye may bedetermined under one or more associated viewing condition, and theprescription can be calculated so that the eye has appropriate powerssuitable for viewing at each additional viewing condition.

The prescription may be derived by determining at least one coefficientof a set of Zernike polynomials. Calculating the prescription oftencomprises determining a plurality of selected Zernike coefficients ofspherical aberration at various orders. The eye at the first viewingcondition may be viewing at a first viewing distance, and the eye at thesecond viewing condition may be viewing at a second viewing distancewhich is less than the first distance, with the second power being morenegative than the first power. The eye at the first viewing conditioncan have a power between 0.25 D and −0.25 D, and the eye at the secondviewing condition may have a power between −0.5 D and −3.0 D.

In another aspect, the invention may comprise a method for deriving aprescription for an eye. The method comprises determining a polynomialexpansion from a wavefront of an eye, and calculating a plurality ofeffective powers based on a plurality of expansion coefficients of thepolynomial expansion at different viewing pupil sizes. The prescriptionmay be generated so as to provide a plurality of desired effectivepowers at said pupil sizes.

In yet another aspect, the invention provides a method for determiningan effective power of an eye under a viewing condition. The methodcomprises determining a plurality of coefficients of a Zernikepolynomial expansion from a wavefront of an eye while the eye has afirst pupil size, and determining a second pupil size of the pupil underthe viewing condition. The effective power of the eye is calculated fromat least one of the coefficients of the Zernike polynomial from arelationship between effective power and pupil size.

In yet another aspect, the invention provides a system for correctingrefraction of an eye, the system comprising at least one input for afirst pupil size of the eye under a first viewing condition and a secondpupil size of the eye under a second viewing condition. A prescriptioncalculating module calculates a presbyopia-mitigating prescription forthe eye in response to the pupil sizes so that the eye has a first powersuitable for the first viewing condition at the first size and so thatthe eye has a second power suitable for the second viewing condition atthe second size.

In another aspect, the invention provides a system for deriving aprescription for an eye, the system comprising a polynomial expansionmodule having an input for a wavefront of an eye and an output for apolynomial expansion. An effective power module has an input coupled tothe output of the polynomial expansion module and an output. Theeffective power module determines an effective power from the polynomialexpansion. A prescription module is coupled to the effective powermodule. The prescription module generates the prescription so as toprovide a plurality of different desired effective powers at anassociated plurality of different viewing pupil sizes.

In yet another aspect, the invention provides a system for determiningan effective power of an eye under a viewing condition, the systemcomprising a first input for a plurality of coefficients of a Zernikepolynomial expansion from a wavefront of an eye while the eye has afirst pupil size. A second input accepts a second pupil size of thepupil under the viewing condition. An effective power calculating modulecalculates the effective power of the eye from at least one of thecoefficients of the Zernike polynomial and a relationship betweeneffective power and pupil size.

For a fuller understanding of the nature and advantages of the presentinvention, reference should be had to the ensuing detailed descriptiontaken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a relationship between accommodation and pupil sizewhen healthy eyes adjust to differing viewing distances.

FIG. 1B illustrates one exemplary relationship between effective powerof an eye and pupil size for a patient, as can be provided from thepresbyopia prescriptions of the present invention by generating anoptical shape which effects desired changes in power with changes inpupil size of a particular patient under differing viewing conditions.

FIG. 1C illustrates a relationship between manifest power and pupildiameter, for example, as measured from patients having differing pupildiameters who have been successfully treated with apresbyopia-mitigating prescription. Such a relationship may be used toidentify a desired change in optical power with changes in pupildiameter for a specific patient.

FIGS. 2A-2C graphically illustrate optical properties of an eye relevantto presbyopia.

FIG. 3 is a flow chart illustrating exemplary method steps foroptimizing a presbyopia-mitigating optical prescription.

FIG. 4 graphically illustrates two presbyopia-mitigating prescriptionsfor an eye of a particular patient.

FIGS. 5A and 5B illustrate alternative presbyopia-mitigatingprescriptions optimized for an eye of a particular patient, and theircharacteristics.

FIG. 5C illustrates a comparison of optimizer values using even-termpolynomials and all power term polynomials for pupil sizes of 4 mm, 5mm, and 6 mm.

FIGS. 6A-D, show alternative presbyopia-mitigating prescriptionsoptimized for an eye of a particular patient.

FIG. 7 illustrates effects of random noise on presbyopia-mitigatingprescriptions optimized for an eye of a particular patient.

FIGS. 8A-C, compares presbyopia-mitigating optimized prescriptions toalternative treatments for differing pupil sizes.

FIGS. 9A-C, compares presbyopia-mitigating optimized prescriptions toalternative treatments for a range of viewing distances.

FIG. 10 illustrates simulated viewing charts viewed at differingdistances to compare presbyopia-mitigating optimized prescriptions toalternative treatments.

FIGS. 11-13 illustrate graphical interface computer screen displays fora prescription optimizer and system.

FIGS. 14 and 15 illustrate pupil sizes and changes at differing viewingconditions for a particular patient.

FIG. 16 graphically illustrates optimizer values for differing levels ofresidual accommodation.

FIG. 17 illustrates effects of pupil change and residual accommodationon presbyopia-mitigating optimized prescriptions for a particularpatient.

FIGS. 18A-C illustrate effects of pupil change and residualaccommodation on presbyopia-mitigating optimized prescriptions for aparticular patient.

FIGS. 19-21 compare optical properties and results of eyes correctedwith a presbyopia-mitigating optimized prescriptions to alternativetreatments.

FIG. 22 schematically illustrates a system for determining apresbyopia-mitigating prescription for a particular patient anddelivering that treatment using laser refractive surgery.

FIG. 23 schematically illustrates a presbyopia-mitigating shape having acentral add region.

FIGS. 24 and 25 schematically illustrates residual accommodation andpresbyopia treatments for increasing a focal range.

FIGS. 26-32 graphically illustrate results from presbyopia-mitigatingtreatments for a population of individual patients.

FIG. 33 graphically illustrates accommodation through a range ofdiffering patient ages.

FIG. 34 schematically illustrates another system for determining apresbyopia-mitigating prescription for a particular patient anddelivering that treatment using laser refractive surgery.

FIGS. 35 and 36 graphically illustrate a presbyopia-mitigatingprescription derived so as to provide appropriate effective powers attwo differing viewing conditions for a particular patient.

FIGS. 37 and 38 graphically illustrate a presbyopia-mitigatingprescription derived so as to provide appropriate effective powers atthree differing viewing conditions for a particular patient.

FIGS. 39 and 40 graphically illustrate a presbyopia-mitigatingprescription derived so as to provide appropriate effective powers atfour differing viewing conditions for a particular patient.

FIGS. 41A and 41B graphically illustrate different presbyopia-mitigatingprescriptions which provide differing effective power variationcharacteristics during pupil size changes under differing viewingconditions.

FIGS. 42 and 43 graphically illustrate effects of different pupil sizeson derived presbyopia-mitigating prescriptions and their opticalcharacteristics.

FIG. 44 illustrates simulated eye-chart letters as viewed with apresbyopic eye treated with a presbyopia-mitigating prescription derivedfor a particular patient.

FIGS. 45A and 45B illustrate an exemplary power/pupil correlation andcorresponding presbyopia prescription.

DETAILED DESCRIPTION OF THE INVENTION

Although the methods, devices, and systems of the present invention aredescribed primarily in the context of a laser eye surgery system, itshould be understood that the techniques of the present invention may beadapted for use in other eye treatment procedures and systems such ascontact lenses, intra-ocular lenses, radial keratotomy, collagenouscorneal tissue thermal remodeling, removable corneal lens structures,glass spectacles, and the like.

The present invention is useful for enhancing the accuracy and efficacyof photorefractive keratectomy (PRK), laser in situ keratomileusis(LASIK), laser assisted epithelium keratomileusis (LASEK), and the like.The present invention can provide enhanced optical correction approachesby improving the methodology for scaling an optical shape, or bygenerating or deriving new optical shapes, and the like.

The techniques of the present invention can be readily adapted for usewith existing laser systems, including the VISX Excimer laser eyesurgery systems commercially available from VISX of Santa Clara, Calif.By providing improved corneal ablation profiles for treating opticaldefects, the present invention may allow enhanced treatment of patientswho have heretofore presented difficult or complicated treatmentproblems. When used for determining, deriving, and/or optimizingprescriptions for a particular patient, the systems and methods may beimplemented by calculating prescriptions for a range of patients, forexample, by calculating discrete table entries throughout a range ofpatient characteristics, deriving or empirically generating parametricpatient characteristic/prescription correlations, and the like, forsubsequent use in generating patient-specific prescriptions.

Referring first to FIG. 1A, the present invention will often takeadvantage of the fact that the eye changes in two different ways withchanges in viewing distance: the lens changes in shape so as to provideaccommodation, and the pupil size simultaneously varies. Accommodationand pupillary constriction work in unison in normal healthy eyes whenshifting from a far to a near viewing distance, and a fairly linearrelation may exist between at least a portion of the overlappingconstriction and accommodation ranges, but the effect may varysignificantly among subjects (from 0.1 to 1.1 mm per diopter). Moreover,when the stimulus for accommodation is increased beyond the eye'sability to change its refraction, the relationship between accommodationof the lens and pupillary constriction may be curvilinear as shown.

While they work in unison, pupillary constriction and accommodation arenot necessarily linked. These two functions may proceed independently,and may even work in opposite directions, particularly when the patientis simultaneously subjected to large variations in light intensity withchanges in viewing distance. Nonetheless, prescriptions for presbyopiacan take advantage of the correlation between pupil dimension andviewing distance for a particular patient. The effective time span for apresbyopia-mitigating prescription may also be extended by accountingfor gradual changes in pupil dimension over time (such as the gradualshrinkage of the pupil as one ages) with the concurrent gradual decreasein the accommodation. Details regarding constriction of the pupil werepublished in a book entitled The Pupil by Irene E. Loewenfeld (IowaState University Press, 1993).

Referring now to FIGS. 1B and 1C, if we assume that we can tailor abeneficial overall optical power for the eye as it changes to differentpupil sizes, we may first want to identify a relationship between thisdesired optical power and pupil size. To determine what powers would bedesirable for a particular patient at different viewing conditions, wemight measure both the manifest sphere and corresponding pupil sizes ofthat patient at a variety of different viewing conditions. The manifestsphere may then be used as our desired or effective power to be used fortreating presbyopia, as detailed below. The desired optical power mightalso be determined from the measured manifest, for example, with desiredpower being a function of the manifest to adjust for residualaccommodation and/or anticipated aging effects or the like. In eithercase, these patient-specific measurements can be the basis fordetermining desired powers for associated pupil sizes of that patient,such as at the four points illustrated in FIG. 1B. Fewer or more pointsmight also be used.

Alternatively, manifest sphere and pupil size for a population ofdifferent patients who have been successfully treated with a givenpresbyopia prescriptive shape may be plotted, and a correlation derivedfrom this empirical data, as schematically illustrated in FIG. 1C. Stillfurther approaches may be employed, including combinations where apopulation of patients having differing pupil sizes are used to derivean initial correlation, which is subsequently refined with multiplemeasurements from at least one patient (and often a plurality ofpatients). Regardless, the relationship between our desired opticalpower and the pupil size can be determined. As will be clear from thedetailed description below, constriction of the pupil at differingviewing distances then allows the overall power of the eye to be alteredby the pupillary constriction, despite a loss in the flexibility of thelens. For example, we can employ a peripheral portion of the ocularsystem having a different power than a central portion. By understandingthe variations of these often aspherical optical systems with changingpupil sizes, we can provide good optical performance throughout a rangeof viewing distances.

The following description will first provide techniques and devices foriteratively optimizing refraction for treatment of presbyopia. This isfollowed by a brief review of an exemplary initial laser ablation shapefor mitigation of presbyopia, which is in turn followed by anexplanation of techniques for optimizing that shape (or other shapes),often using empirical and/or patient-specific information to scale theshape. Generalized analytical and numerical techniques for determiningor selecting appropriate presbyopia mitigating prescription shapes willthen be provided.

When designing a prescriptive shape for a presbyopia eye treatment, itis useful to select a mathematical gauge of optical quality appropriatefor presbyopia for use as a goal function. This allows forquantification and optimization of the shape, and for comparison amongdifferent shapes. The present invention provides methods forestablishing a customized optical shape for a particular patient basedon a set of patient parameters per the goal function. By incorporatingiterative optimization algorithms, it is also possible to generate ashape having an optimized level of optical quality for the particularpresbyopic patient.

Selecting a Goal Function Appropriate for Presbyopia

The goal function relates to optical quality, and it can be, forexample, based on, or a function of (or related to) optical metrics suchas Strehl ratio (SR), modulation transfer function (MTF), point spreadfunction (PSF), encircled energy (EE), MTF volume or volume under MTFsurface (MTFV), or contrast sensitivity (CS); and optionally to newoptical metrics which are appropriate to presbyopia, such as compoundmodulation transfer function (CMTF) as described below. In opticalterms, the goal function should make sense. That is to say, minimizationor maximization of the goal function should give a predictable optimizedoptical quality of the eye. The goal function can be a function with acertain number of free parameters to be optimized (minimized) through anoptimization, or minimization, algorithm.

Although there are many types of goal functions available for use withthe present invention, the discussion below generally touches on twobroad schools of goal functions. In a Diffraction Theory based approach,the shape is considered as a wave aberration. Typically, a Fouriertransform is employed for calculating optical quality relatedparameters, such as Strehl ratio (SR), modulation transfer function(MTF), MTF volume or volume under MTF surface (MTFV), compoundmodulation transfer function (CMTF), or contrast sensitivity (CS),encircled energy (EE) (based on point spread function), as well asspecial cases that combine one or more of these parameters, or values ofthe parameters in specific situations (such as MTF at spatial frequencyor encircled energy at a field of view), or integration of anyparameters (volume of MTF surface at all frequencies or up to a cutofffrequency, for example 60 cycles/degree or 75 cycles/degree, because 60cycles/degree is the retina cone's limiting spatial frequency). In aGeometrical Optics approach, or the so-called ray tracing approach, theoptical effect is based on ray tracing. With both the Diffraction Theoryand the Geometrical Optics approaches, polychromatic point spreadfunction with Stiles-Crawford effect, chromatic aberrations as well asretina spectral response function can be used.

Monochromatic point spread function (PSF) has been used for describingoptical defects of optical systems having aberrations. Due to the simplerelationship between wave aberrations and the PSF for incoherent lightsource, Fourier transform of the generalized pupil function has beenused in the calculation of point spread functions. Most opticalapplications, however, do not use a monochromatic light source. In thecase of human vision, the source is essentially white light. Thus, thereare limitations associated with the use of monochromatic PSF as a goalfunction.

Polychromatic point spread function (PSF) with correct chromaticaberrations, Stiles-Crawford effect as well as retina response function,can be used for optical modeling of human eyes. Here, chromaticaberrations are due to the fact that light from different wavelengthwill focus either in front of the retina or behind it. Only portions ofthe light will focus exactly on the retina. This gives the eye anextended depth-of-focus, i.e., if one has focusing error of some amount,the eye is still capable of focusing at least for some wavelengths.Therefore, chromatic aberrations in fact help the correction ofpresbyopia. If the depth-of-focus is sufficiently large, there would beno presbyopia problem. Unfortunately, the chromatic aberrations are notlarge enough and it also varies with the wavelength. Stiles-Crawfordeffect, also known as pupil apodization, is due to the waveguideproperty of the retinal cones. Light from the pupil periphery has aslightly less chance of being detected by the retina because the ray oflight might not reach the bottom of the cone, due to a slight incidentangle. As for the retinal spectral response function, it is known thatthe cones, which are responsible for daylight vision, have differentsensitivity to different wavelengths. Only green light is absorbed bythe eye almost completely. Both blue light and red light are absorbed bythe eye partially.

Once the PSF is calculated, calculation of the Strehl ratio isstraightforward. Strehl ratio can be defined as the ratio of the peak ofthe point spread function (PSF) of an optical system to the peak of adiffraction-limited optical system with the same aperture size. Anexample of a Strehl ratio is shown in FIG. 2A. A diffraction-limitedoptical system is typically a system with no aberrations, or opticalerrors. It can be an ideal or perfect optical system, having a Strehlratio of 1.

The goal function can also be a function of modulation transfer function(MTF). Modulation transfer function can be used to predict visualperformance. Typically, the MTF at one spatial frequency corresponds toone angular extent of targets. The modulation transfer function (MTF)can be calculated with the following formulations:MTF(u,v)=FT[PSF(x,y)]MTF(u,v)=Re[GPF(x,y)

GPF(x,y)]where u and v represent spatial frequencies, Re represents the real partof a complex number, FT represents a Fourier Transform, GPF represents ageneralized pupil function, and x and y represent position or field ofview. An example of an MTF is shown in FIG. 2B.

Another example of a goal function, the compound MTF, can be defined asF(v)=(α₁ MTF ₁+α₂ MTF ₂+α₃ MTF ₃)/3where MTF₁, MTF₂, and MTF₃ are the MTF values at 10 cycles/degree, 20cycles/degree and 30 cycles/degree, respectively. These correspond toSnellen eye chart of 20/60, 20/40 and 20/20 visions, respectively. Theweighting coefficients α₁, α₂, α₃ can be chosen so that 1/α₁, 1/α₂, 1/α₃are the diffraction-limited MTF at these spatial frequencies,respectively. Therefore, in the diffraction-limited case, the compoundMTF F(v) can have a maximal value of unity.

Where MTF at one spatial frequency corresponds to one angular extent oftargets, compound MTF can be defined as linear combination of MTF atdifferent spatial frequencies normalized by a diffraction-limited MTF,and can similarly be used to predict visual outcome. A more generalformula for CMTF is

${{CMTF}(v)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}{\alpha_{i}{{MTF}_{i}(v)}}}}$where α₁ is the reciprocal of the i-th diffraction-limited MTF. In somecases, three MTF curves at 10, 20 and 30 cycles per degree are used. Anideal value of CMTF can be about 1. Good values can be about 0.2 orabout 0.3. In a healthy eye, the spatial frequency limit can be about 60cycles per degree due to the configuration of retina cones. In thetreatment of presbyopia, however, it may not be necessary to provide atreatment corresponding to this limit, as the treatment will ofteninvolve a compromise of good distance and near sight. Optionally, aminimum distance vision gauge desired target may be provided, with nearsight being optimized and, as needed, compromised.

FIG. 2C shows an example of the compound MTF over a vergence of 3diopters (upper panel) versus its corresponding individual MTF curves at10, 20, and 30 cpd (cycles per degree). Compound MTF can correlate wellwith visual acuity and contrast sensitivity at the same time, at leastoptically.

To establish an optically optimized shape appropriate for presbyopia, atleast one of the goal functions, such as Strehl ratio, encircled energy,or MTF, MTF volume or volume under MTF surface (MTFV), compoundmodulation transfer function (CMTF), or contrast sensitivity (CS) shouldbe maximized. For improved presbyopia treatment, the optical metric canbe maximized in all target vergence, that is, for targets at alldistances. Furthermore, it is also desirable to minimize the fluctuationof the goal function. Therefore, the goal function, which isincorporated into the optimization algorithm of the optimizer, can bedefined as

where O is the goal function; c₁, c₂, . . . are the polynomialcoefficients; PAR is presbyopia-add to pupil ratio (described below); vis the vergence; F(v) is one of the optical metrics; σ is the standarddeviation of F(v), PV is the peak-to-valley of F(v); and v₀ is the endpoint of the vergence range, which may be (for example) between 15 and100 cm, such as 40 cm. Because ∫dv is a constant, either a smaller a ora larger ∫F(v) dv can minimize the goal function O.

The formulas given here are examples of the many formulae that can beused as the goal function. The basic approach will often be to provide agoal function that is optimized to give as practical a solution aspossible for presbyopia correction.

Selecting an Iterative Optimization Algorithm

Any of a number of optimization algorithms may be used by the optimizerto maximize, minimize, or otherwise globally or locally optimize thegoal function. Because many numerical algorithms use functionminimization concept, it is often convenient, but not necessarilyrequired, to use minimization of the goal function. As examples,N-dimensional minimization algorithms such as the Downhill Simplexmethod, the Direction Set method, and the Simulated Annealing method canbe used to optimize the goal function. Likewise, the algorithm describedby Press et al., in “Numerical Recipes in C++”, Cambridge UniversityPress, 2002 can also be used. Algorithms such as those listed above areoften used for function optimization in multi-dimensional space.

The Downhill Simplex method starts with an initialization of N+1 pointsor vertices to construct a simplex for an N-dimensional search, and inevery attempt tries to reflect, stretch, or shrink the simplex bygeometrical transformation so that a close-to-global minimum orpre-defined accuracy can be found. When Gaussian random noise ofstandard deviation of 0.02 μm in optical path difference (OPD) is added,the algorithm still converges, with no degradation.

In the case of Direction Set method, also known as Powell's method, None-dimensional vectors are initialized and the N-dimensional search issplit in such a way that a one N-dimensional vector is chosen and theminimization is done in that direction while other variables (N−1dimensions) are fixed. This process is continued until all dimensionsare covered. A new iteration is initiated until the pre-determinedcriterion is met. The Direction Set method can use a separateone-dimensional minimization algorithm such as a Golden section search.

The Simulated Annealing method, which is useful for dealing with a largenumber of uncertainties, starts with an initial configuration. Theobjective is to minimize E (analog to energy) given the controlparameter T (analog to temperature). Simulated Annealing is analogous toannealing, is a recent, proven method to solve otherwise intractableproblems, and may be used to solve the ablation equation in laserablation problem. This is more fully described in PCT Application No.PCT/US01/08337, filed Mar. 14, 2001, the entire disclosure of which isincorporated herein by reference. Simulated annealing is a method thatcan be used for minimizing (or maximizing) the parameters of a function.It is particularly suited to problems with very large, poorly behavedfunction spaces. Simulated annealing can be applied in the same wayregardless of how many dimensions are present in the search space. Itcan be used to optimize any conditions that can be expressednumerically, and it does not require a derivative. It can also providean accurate overall minimum despite local minima in the search space,for example.

FIG. 3 shows the flow chart of an overall method for shape optimizationfor presbyopia correction. Each functional block may contain one or morealternatives. To create a presbyopia add-on shape W(r), an iterativefunction minimization algorithm can be employed such that the goalfunction, which could be a function of any suitable optical metrics(e.g. SR, MTF, EE, CMTF, MTFV, CS) is itself optimized to solve for anunknown shape. The shape can be expanded into a set of even power termpolynomials (EPTP) or non-EPTP (i.e. all power term polynomials). EPTPrefers to polynomials that only have the even power terms, for instance,F(r)=ar²+br⁴+cr⁶. The goal function should have good correlation withvisual performance, at least optically. Point spread function can becalculated to obtain additional and/or alternative optical metrics. Thepresbyopia prescription can refer to an optical surface that can be usedto treat or mitigate presbyopia. It can correspond to, for example, theshape of a spectacle lens, a contact lens, an intra-ocular lens, atissue ablation profile for refractive surgery, and the like.

It is desirable that the optimizer provide satisfactory outcome in termsof attributes such as result, convergence, and speed. FIG. 4 shows acomparison of Direction Set method and Downhill Simplex method for thefollowing inputs: pupil size 5.6 mm, vergence 3 D and vergence step 0.1D. Direction Set method uses 17 iterations and Downhill Simplex methoduses 152 iterations. Each Direction Set method iteration takes longerthan each Downhill Simplex method iteration. The optimizer value for theDirection Set method is 2.8 while that for the Downhill Simplex methodis 2.658. Shape for left panel is as−0.9055r²+6.4188r⁴−2.6767r⁶+0.5625r⁸ with ratio of 0.7418.

Both algorithms seem to converge to a similar shape, although the depthsof the shapes are different. Considering the difference in the pupilratio, however, the actual shapes within 70% of the pupil radius arequite close. When the vergence step is smaller, each iteration can takea longer time, but the overall number of iterations often tends tobecome smaller.

Inputting an Initial Prescription Into an Optimizer

The initial prescription, often comprising an optical surface shape, maybe defined by an expansion such as a polynomial (EPTP, non-EPTP), aZernike polynomial, a Fourier series, or a discrete shape entirety. Adiscrete shape entirety can also be referred to as a direct surfacerepresentation by numerical grid values. The prescription shape may beassumed to be circularly or radially symmetric, with the aim ofapproaching an emmetropic eye. The symmetric shape can be decomposedinto a set of polynomials, such that it has one or more independentvariables. One of the variables can be the presbyopia-add to pupil ratio(PAR), or the ratio of the shape diameter to the pupil diameter. When acentral power add region is employed (as described below), the PAR canbe the ratio of the radius of the presbyopia-add to the radius of thepupil. It will also be appreciated that the ratios discussed herein canbe based on area ratios or on diameter or radius ratios. It should beassumed that when diameter or radius ratios are discussed, thatdiscussion also contemplates area ratios. In certain cases, the PAR canrange from about 0.2 to about 1.0. Relatedly, in some cases the methodsof the present invention can constrain the PAR to range from about 0.2to about 1.0. The other variables can be the coefficients of eachpolynomial term. For example,Shape(r)=ar+br ² +cr ³ +dr ⁴ +er ⁵ +fr ⁶

The diameter of the shape can be larger than the pupil size, but if sospecial considerations may need to be taken. For example, it may benecessary to only consider the net shape within the pupil.

The polynomials can be normal polynomials or polynomials with even powerterms only. For example, even-power-term polynomials (EPTP) up to the6^(th) or 8^(th) order can be used to obtain a practically good output,that is, a practical optimal shape for the particular patient. Residualaccommodation can also play an active role in presbyopia correction. Ina related instance, normal presbyopes can be treated with theprescription obtained in this approach together with a prescription forthe correction of the refractive error.

As an example, a circularly or radially symmetric, pupil-size dependentshape for presbyopia-add can be assumed for emmetropic presbyopes. Theshape can then be expanded to polynomials up to the 6^(th) or 8^(th)order. With the optimization procedure, it is found that polynomialexpansion of the shape up to the 6^(th) or 8^(th) order can be used toobtain a practical optimal shape for presbyopia correction.

In a wavefront with aberrations, denoted by W(r, θ), the wavefront canbe thought of as an optimal shape for presbyopia correction. Thepolychromatic PSF can be expressed as

${PSF} = {\sum\limits_{\lambda}{{R(\lambda)}{{{FFT}\left( {{P_{sc}(r)}{\exp\left\lbrack {{- j}{\frac{2\pi}{\lambda}\left\lbrack {{W\left( {r,\theta} \right)} + {\alpha\;{D(\lambda)}} + {V(l)} + {{RA}(l)}} \right\rbrack}} \right)}} \right.}^{2}}}}$where R(λ) is the retina spectral response function and can beapproximated toR(λ)=e^(−300(λ-λ) ₀ ⁾ ²and P(r) is the pupil apodization function (Stiles-Crawford) and can bewritten as

${P_{sc}(r)} = 10^{{- \rho}\;\frac{r^{2}}{R^{2}}}$and D(λ) is chromatic aberration at wavelength λ and is close toD(λ)=−21.587+92.87λ−134.98λ²+67.407λ³and V(l) is the vergence induced aberration at distance l meters, andRA(l) is the residual accommodation induced aberrations with a differentsign as compared to V(l). When there are no aberrations, RA(l) cancancel V(l) as long as there is enough residual accommodation in theeye. Here, the central wavelength λ is taken as 0.55 μm (as allwavelength units in the above formulae are in μm). The pupil apodizationstrength parameter ρ is taken as 0.06. α is the conversion factor fromdiopter to optical path difference (OPD). FFT denotes a fast Fouriertransform and |*| denotes the module of a complex number.

The polychromatic point spread function, or PPSF, can be the pointspread function of an eye as calculated with consideration of thepolychromatic nature of the incident light. Further, the chromaticaberrations, the Stiles-Crawford effect, as well as the retinal spectralresponse function can also be considered.

The vergence induced aberration, or VIA, can be equal to the reciprocalof the vergence distance. When a target at a certain distance is viewedby the eye, it is the same as viewing the target at infinity but the eyehas an additional aberration, the vergence induced aberration.

For emmetropic eyes, it may be desirable that the wavefront that isoptimized be circularly symmetric. Therefore, it can be decomposed intoa set of polynomials (non-EPTP) asW(r)=ar+br ² +cr ³ +dr ⁴ +er ⁵+ . . .

However, if it is desirable that the edge of the shape be smoother, itmay be advantageous to decompose the wavefront into a set ofeven-power-term polynomials (EPTP) asW(r)=ar ² +br ⁴ +cr ⁶ +dr ⁸+ . . .

Using even power term polynomials (EPTP) also can help to establish asurface shape that is more round at the center, which creates certainmanufacturing or ablation efficiencies.

It may also be useful to denote another parameter, t, to be the ratio ofthe radius of the wavefront R to the radius of the pupil R₀. This isbecause both D(λ) and V(l) can have the same size as the pupil and W(r)usually has a smaller size. When the calculated t is larger than 1, theshape can become larger than the pupil. In this case, only the portionof the shape up to the pupil size is used for optical qualityevaluation.

As depicted in FIG. 5A, although normal polynomials can give slightlybetter optimizer values than even-power-term polynomials, theprescription may be harder to realize. FIG. 5A illustrates a comparisonof shapes with normal polynomials (left panel) and with even-power-termpolynomials (right panel). The shape on the right panel can be expandedas −1.6154r+1.7646r²+1.2646r³+1.9232r⁴+0.1440r⁵+0.1619r⁶ with a ratio of0.8 and the shape on the left panel can be expanded as−1.1003r²+8.2830r⁴+0.7305r⁶−2.2140r⁸ with a ratio of 0.9106. Both weredetermined using Downhill Simplex method for a pupil size of 5.6 mm andvergence of 3 D with 0.1 D step, without residual accommodation. Theleft panel shows an optimal shape for 6 normal polynomial terms and theright panel shows an optimal shape with 4 EPTP terms. It has been foundthat polynomials up to the 8^(th) power (4 EPTP terms) appear to givehighly satisfactory results.

FIG. 5B shows another comparison of EPTP and non-EPTP expansions. Theleft panel shows an optimized shape based on an 8th order expansion(EPTP), whereas the right panel shows an optimized shape based on a 3rdorder expansion (non-EPTP). In general, shapes derived from an EPTP havea smoother shape with a flat central zone. This flat central zone cancorrespond to good distance visual performance.

Another comparison of EPTP and non-EPTP expansions is provided in FIG.5C, which shows optimized (minimized) values with EPTP and non-EPTPexpansion for a 4, 5, and 6 mm pupil over a 3 D vergence distance. Ingeneral, non-EPTP optimization gives a slightly smaller (more optimized)value than EPTP. Sixth-order EPTP appears to give the smallest value for4 mm and 5 mm pupils and eighth-order EPTP appears to give the smallestvalue for a 6 mm pupil. Third-order non-EPTP appears to give thesmallest value for 4 mm and 5 mm pupils and fourth-order non-EPTPappears to give the smallest value for a 6 mm pupil.

Using an even-power-term polynomial (EPTP) expansion can result in asmoother shape than a non-EPTP expansion. This smooth shape can be theminimal requirement for good distance vision. In general, 6^(th)-orderor 8^(th)-order EPTP expansion and 3^(rd)-order or 4th-order non-EPTPexpansion result in good optimized value. Without residualaccommodation, larger pupils can be more difficult to optimize thansmaller pupils. This is shown, for example, in FIG. 8A.

The optimized multi-focal shape appears to give much more balancedresults for the correction of presbyopia than bi-focal and multi-focalshapes.

In addition to using a general polynomial expansion for the optimalsurface, it is also possible to use other means of surface expansion.For example, Zernike polynomial expansions may be used. The followingformula presents an example of a Zernike polynomial expansion

${W(r)} = {\sum\limits_{i = 1}^{n}{c_{i}{Z_{i}\left( {r,\theta} \right)}}}$where radially symmetric terms such as Z₄, Z₁₂, and Z₂₄ can be used, andc_(i) are free parameters.

Another way of surface expansion is by means of spectral expansion, orFourier expansion. The following formula presents an example of aFourier expansion.

${W(r)} = {\sum\limits_{i = 1}^{n}{c_{i}{F_{i}(r)}}}$where c_(i) are free parameters. Fourier expansion is based on thepremise that any surface can be decomposed into a set of sinusoidalharmonics with different spatial frequencies. It may not be necessary toexpand the surface to very high spatial frequencies.

Discrete surface, or discrete shape entirety, is another type ofexpansion that can be used in the present invention. Discrete surfacecan be represented by the following formulaW(r)=W _(ij),(i=1,2, . . . , M; j=1,2, . . . , M)where W_(ij) are free parameters (M×M).

Inputting A Set of Patient Parameters into an Optimizer

The set of patient parameters can also be referred to as the set of userinput parameters. The input parameters may provide certain patientcharacteristics, such as pupil size and its variations, desired power,and residual accommodation which can be modeled by factors such asgender, age, and race, or which can be measured by instruments.

Residual accommodation can be measured in diopters. Vergence can also bemeasured in diopters and typically is inversely related to distance,such that a distance of infinity corresponds to a vergence of zero.Similarly, a normal reading distance of ⅓ meters can correspond to avergence of 3 diopters, and a farther distance of 10 meters cancorrespond to 0.1 diopters.

It can be useful to model the residual accommodation in the optimizationprocess. The visual quality of the shape can be optimized given acertain set of conditions such as vergence, residual accommodation, andchromatic aberrations. However, even without a direct correlationbetween optical surface and the visual quality, it may be convenient touse the minimum root-mean-squares (RMS) error to determine theaccommodation during different visual vergence. For instance, if noaberrations are present, and there is 2 D of residual accommodation,such a patient uses 0.5 D of residual accommodation when visualizing atarget at 2 meters. Relatedly, the patient uses all 2 D of residualaccommodation to view a target at 0.5 meters. The patient would havedifficulty viewing targets closer than 0.5 meters, as the residualaccommodation is exhausted and no longer available. People with largerpupils or smaller residual accommodation may be harder to treat.

When aberrations or additional add-on shapes are present, the amount ofresidual accommodation for different visual vergence may vary. Forexample, in a patient having 0.5 D residual accommodation, with anadd-on shape of exactly 1 D added to the eye, the eye may not need toaccommodate until viewing a target at a distance of one meter. Here, the1 D add-on can cover the first diopter of visual vergence, eitherentirely or partially. At a large distance, the visual quality may beworse because the eye cannot accommodate in the reverse direction. Thetechniques of the present invention can be adapted to enhance anoptimizer value at low vergence when residual accommodation is assumed.

When a more complicated add-on shape is used, one way to determine theaccommodation is to calculate the available residual accommodation whichwould minimize the overall RMS.

Shape optimization can be customized for a patient. The customizationcan include the patient's pupil sizes at different lighting and viewingconditions, such as bright far viewing, bright near viewing, dim farviewing, and dim near viewing. The optimization can also be based on thepatient's residual accommodation, or predicted residual accommodationbased on the patient's age, or the patient's vision preference due tofor example, their employment or other requirements. That is to say, thecustomization can put more emphasis on far, near, or intermediateviewing. Similarly, the customization can put more emphasis on dimlighting condition, bright lighting condition or scotopic lightingcondition. Further, the optimization can be based on how long thepatient wishes to have the correction last. In many ways, presbyopiacorrection can be a management of compromise. If a patient needs to haveexcellent correction, he or she might need re-treatment after a coupleof years as he or she gets older, when residual accommodation diminishesand/or the pupil size becomes smaller.

Inputting a Set of Initial Conditions into an Optimizer

The output result, or optical surface shape, can be sensitive to thechoice of the initial condition. In the case of Downhill Simplex method,the initial condition can be the initial N+1 vertices as well as thecorresponding initial optimizer values for an N-dimensional problem. Inother words, the conditions can be the initial vertices, as well as thevalue associated with these vertices, for N independent variables. Inthe case of the Direction Set method, the initial condition can be theinitial N direction's unit vector and an initial point for anN-dimensional problem.

When both or either the initial values for the polynomial coefficientsand the pupil ratio are set low, the resulting actual numbers may oftenbe low, especially for the case of pupil ratio. In one example, theinitial condition is chosen to be 1.75 for all coefficients and 0.26 forpupil ratio. FIGS. 6A-6D show a variety of shapes determined usingdifferent initial conditions, as calculated by the Downhill Simplexmethod. Pupil size of 5.6 mm and vergence of 3 D with 0.1 D step areassumed. Shape for FIG. 6A is 4.12r−0.235r²+0.08r³−6.9r⁴+4.81r⁵+2.157r⁶;for FIG. 6B it is 2.6165r²+4.1865r⁴+6.9123r⁶−9.6363r⁸; for FIG. 6C it is1.7926r+5.0812r²-2.163r³−2.3766r⁴−1.1226r⁵1.6845r⁶; and for FIG. 6D itis −1.5178r²+7.2303r⁴−2.4842r⁶−1.7458r⁸+1.8996r¹⁰.

For the initial conditions, totally random input and fixed ratios maynot necessarily help the algorithm to converge to a global minimum ormaximum.

Implementing an Optimizer to Establish a Customized Optical Shape forthe Particular Patient Per the Goal Function so as to Treat or MitigatePresbyopia in the Particular Patient

The iterative optimization algorithm can be employed to calculate ashape that optimizes the optical quality for the particular patient fordistance vision and near vision. In other words, the corrective opticalsurface shape corresponds to the set of output parameters provided bythe optimizer. The output parameters are the coefficients of polynomialsdescribing the shape, as well as the ratio of diameter of the shape tothat of the pupil diameter. These output parameters can define the finalcustomized or optimized optical surface shape. This approach provides anumerical way for general optimization of the optical surface shape forpresbyopia correction. Whether it is for refractive surgery, contactlens, spectacle lens, or intra-ocular lens, the approach can be verybeneficial. For presbyopes with refractive error, the optimal shape canbe combined with the shape that corrects for the refractive error, forexample the patient's measured wavefront error.

In order to model such deviation in practice, Gaussian distributed noisecan be added into the shape so that when noise is present the stabilityof the algorithm can be tested. For example, Gaussian noise of standarddeviation of 0.02 μm OPD can be introduced. This corresponds to nearly0.06 μm in tissue depth in the case of laser surgery. This is largerthan the general RMS threshold for the Variable Spot Scanning (VSS)algorithm for such a shape. FIG. 7 illustrates a comparison of theshapes calculated with a noise-free (dark) condition and with a 0.02 μmstandard deviation of Gaussian random noise in OPD on the wavefront. Thenoise-free case has an optimizer value of 3.008 with 184 iterations andthe noisy case has an optimizer value of 2.9449 with 5000 iterations.Both use Downhill simplex method. Pupil size is 5 mm with 3 D vergenceand 0.1 D step. Noise addition can also help to guarantee the stabilityof the algorithm.

It is also possible to test how the convergence, optimizer value, andshape work with different input pupil sizes. An example of results fromsuch a test is shown in Table 1. For smaller pupil sizes, the shape cancover the whole pupil. That is to say, the shape can be larger than thepupil size. Also, the depth may tend to become smaller with smallerpupils.

TABLE 1 Shapes for pupil dependency with 3 D vergence and 0.1 D step. #Pupil Iterations A B C D T Value Depth 6.0 234 −1.5688 12.0893 −0.5895−2.6934 0.9866 2.6808 7.2881 5.8 316 −0.5212 4.4186 −0.8472 −0.07640.6870 2.8215 2.9980 5.6 152 −1.1003 8.2830 0.7305 −2.2140 0.9106 2.65805.7356 5.4 274 −0.5918 5.0881 1.2448 −1.1930 0.9124 2.7539 4.5651 5.2269 −1.4101 5.3067 −0.4326 −0.4379 0.7944 2.7979 3.1210 5.0 186 0.40792.2298 0.0598 1.1958 0.9446 3.0080 3.8933 4.8 531 −3.4870 54.962548.5083 −125.31 1.8427 2.6772 4.0692 4.6 492 −1.3517 8.5336 −4.81381.6981 0.999 2.5871 4.1223 4.4 422 −2.1972 17.2673 32.1306 −44.9031.5095 2.6924 3.4652 4.2 163 −0.8345 4.2663 4.3575 −3.5136 1.1093 2.71962.9770 4.0 545 −4.8205 29.1525 7.9952 −23.086 1.5984 2.6822 2.7003 3.8333 0.1519 0.6105 2.5097 −1.6318 0.7765 3.0533 1.6403 3.6 177 −1.04221.4185 2.2061 −0.9600 0.9736 2.7533 1.7636 3.4 230 −3.6844 19.08784.2289 5.3957 1.6909 2.7202 1.4760 3.2 219 −1.2266 1.9391 0.8145 0.29141.0989 3.0486 1.0858 3.0 287 3.3482 −2.5793 0.8977 −0.3937 0.9941 2.90611.3286 2.8 257 −0.2052 0.2657 0.0451 0.2494 0.7920 2.8933 0.3890 2.6 136−0.6749 1.8074 0.3418 −0.3918 1.0637 2.7377 0.8731 2.4 332 −2.845516.408 −13.119 0.9270 1.5988 3.0920 0.7653 2.2 239 −2.6435 2.2329 1.9556−1.7776 0.8557 3.1667 0.6329 2.0 303 −0.6398 0.9010 0.5835 −0.36010.9527 3.9384 0.5827

As determined by the approach of the present invention, one desirableoptical surface shape has a central un-ablated zone and an outside zonethat provides improved near vision or reading capability. Based on theexample shown in FIG. 4, the central flat zone can be about 1.96 mm indiameter. Because the healing effect may reduce the central zone, theplanned flat ablation may need to go beyond 2 mm in order to get ahealed flat zone of about 1.96 mm. This can be for a pupil size of about5.6 mm (natural size). The present invention can also consider practicalpupil dependency in the approach. In one example of the presentinvention, the optical zone can go to about 0.91 times the size of thepupil size, which is about 5.1 mm. Further, the present invention mayalso incorporate a transition zone such as the VISX standard transitionzone technique, as used in variable spot scanning (VSS). What is more,the present invention can also provide a clear mathematical descriptionfor the optical surface shape outside of the un-ablated zone.

Relatedly, FIG. 8C illustrates that there can be a dependency betweenoptimizer value and pupil size. FIG. 8C also shows a preferred optimizervalue (optimal). An optimizer value can be a value of the goal functionafter it is optimized. Theoretically, this value should not be smallerthan unity. An optimization, or minimization, algorithm can be used tofind values of free parameters such that the optimizer value is as closeto unity as possible.

The present invention can incorporate varying pupil sizes, althoughpresbyopes may tend to have smaller pupil size variation. Because anoptimal shape for a fixed pupil size may no longer be optimized if thepupil size changes, the present invention can provide approaches thatcan allow for pupil size variations. The final optical surface shape canbe one that gives an optimal optical quality over a certain vergencerange when the pupil size varies over a range.

To demonstrate how effective a solution is in terms of optical metrics,the MTF can be shown at different spatial frequencies, as illustrated inFIGS. 8A-C, which provides optimizer values for various corrections.Apparently the optimal curve gives the minimum (optimized) value for allpupil sizes. Eyes with larger pupils can be more difficult to optimize.What is more, carefully designed multi-focal correction can be close tooptimal, as further illustrated in FIGS. 8A-C. That is, the optimizervalue for the multi-focal correction can be close to that of theoptimized correction, hence the results are quite similar. This outcomeis also illustrated in FIG. 10. The lower regression line in FIG. 8C canset the practical limit for the optimizer value.

In another approach, to demonstrate how effective a solution is in termsof optical metrics, the compound MTF can be plotted, as shown in FIGS.9A-B. Here, the compound MTF for various treatments for a 5 mm pupilover a 3 D vergence is plotted. It can be beneficial to optimallybalance the level of compound MTF at every vergence distance or over thedesired vergence. FIG. 9C shows a comparison of bi-focal and optimalcorrections, with a simulated eye chart seen at different targetdistances, assuming a 5 mm pupil with no accommodation. The eye charthas 2/100, 20/80, 20/60, 20/40, and 20/20 lines, respectively.

FIG. 10 is a simulated eye chart seen at different target distances, andcompares an optimized case (bottom) to no correction (top line); readingglasses (second line); bi-focal lenses (inner half for reading and outerhalf for distance, third line); and multi-focal lenses (pupil center forreading with maximum power and pupil periphery for distance with zeropower and linear power change in between, four line). The effects of theoptimization can be clearly seen by the comparison. No accommodation orrefractive error is assumed in any of the cases. The eye chart has20/100, 20/80, 20/60, 20/40, and 20/20 lines.

Using the above approaches, it is possible to obtain a shape that is notonly larger than the pupil size, but that can also be practicallyimplemented. Often, only the portion of the shape inside the pupil maybe evaluated for optical quality, although this is not a requirement.For example, the entire zone over the pupil can remain un-ablated, butthere may be a zone outside the pupil that is ablated. In this way,distance vision is not affected, but for near vision, there can be anadvantage from light coming outside of the pupil due to greatly deformedperiphery. A goal function based on geometrical optics, or ray tracing,can be useful to determine such shapes.

Residual accommodation can also affect the optimization result, becauseit can remove some of the ripples on the combined wavefront at anyvergence.

The approaches of the present invention can be implemented on a varietyof computer systems, including those with a 200 MHz CPU with 64 MBmemory, and typically will be coded in a computer language such as C orC++. Simulations have successfully been run on a laptop computer with a1.2 GHz CPU with 256 MB memory. The techniques of the present inventioncan also be implemented on faster and more robust computer systems.

The present invention includes software that implements the optimizerfor practical applications in a clinical setting. The optimizer willoften comprise an optimizer program code embodied in a machine-readablemedium, and may optionally comprise a software module, and/or acombination of software and hardware. As shown in FIGS. 11-13, thesoftware interface can comprise two primary panels: the parameter paneland the display panel. The parameter panel can be split into twosub-panels: optimization and verification. The display panel can also besplit into two sub-panels: graph panel and image panel. The software canalso include a menu bar, a tool bar, and a status bar. In the tool bar,small icons can be used for easy access of actions.

The optimization sub-panel can include a number of parameter units. Forexample, a first parameter unit can be the pupil information group. Inthe examples shown in FIGS. 11-13, the user or operator can give fourdifferent pupil sizes for a specific eye. More particularly, the pupilinformation group includes the pupil size in (a) bright distance viewingcondition, (b) bright near viewing condition (e.g. reading), (c) dimlight distance viewing condition, and (d) dim light near viewingcondition (e.g. reading). These different pupil sizes can be used in theoptimization process.

A second parameter unit in the optimization sub-panel can be the displaygroup. In the examples shown in FIGS. 11-13, the user or operator hasthree different choices for the display, including (a) none, (b) shape,and (c) metric. The display group can provide instruction to thesoftware regarding what kind of display is desired for each iteration.For instance, none can mean no display, shape can mean displaying thecurrent shape, and metric can mean displaying the current optical metriccurve over the desired vergence for this current shape. The choices canbe changed during the optimization procedure, and in this sense it isinteractive.

A third parameter unit in the optimization sub-panel can be the opticalmetric group. In the examples shown in FIGS. 11-13, the user has fivedifferent choices for the metric, including (a) Strehl ratio, (b) MTF ata desired spatial frequency, (c) encircled energy at a desired field ofview, (d) compound MTF (CMTF) with a set of specific combinations, whichcould be any number of MTF curves at different spatial frequencies, andwhen the “auto” check box is checked, it can use a default CMTF withthree frequencies, such as, for example: 10, 20 and 30 cycles/degree,and (e) the MTF volume up to a specific spatial frequency. 25% CMTF overthe vergence appears to be an example of a good target value foroptimization.

A fourth parameter unit in the optimization sub-panel can be theoptimization algorithm group. In the examples shown in FIGS. 11-13, theuser has three different choices for the optimization algorithm employedby the optimizer, including (a) the Direction Set (Powell's) method, (b)the Downhill Simplex method, and (c) the Simulated Annealing method. Theoptimizer can employ a standard or derived algorithm for functionoptimization (minimization or maximization). It can be amulti-dimensional, non-linear, and iterative algorithm.

A number of other parameters can be included in the optimizationsub-panel. As shown in FIGS. 11-13, these other parameters can beimplemented separately (optionally as a ComboBox) with a number ofchoices for each. These can include parameters such as (a) the number ofterms of the polynomial expansion, (b) the frame size, (c) the PSF type(monochromatic, RGB, or polychromatic), (d) whether the shape is EPTP ornon-EPTP, (e) the vergence requirement, (f) the vergence step, and (g)the residual accommodation. The software can include a StringGrid tablethat displays the polynomial coefficients, the PAR value, the optimizervalue, as well as the current number of iterations. These numbers can beupdated every iteration.

The verification sub-panel can include a number of parameter units. Forexample, a first parameter unit can be the “which” group. In theexamples shown in FIGS. 11-13, the operator can use this group to selectwhether to use built-in eye chart letters, or an entire eye chart or ascene. A second parameter unit in the verification sub-panel can be theleft image group. The user can make a selection in the left image groupfrom PSF and imported scene. A third parameter unit is the right imagegroup, wherein the user can make a selection from imported scene, andblur at near. The two image display groups are for the left and rightsubpanels in the image subpanel.

As further illustrated in FIGS. 11-13, the ComboBox for letter canprovide a list of different eye chart letters, and the VA ComboBox canprovide the expected visual acuity, from 20/12 to 20/250. The ContrastComboBox can provide a list of contrast sensitivity selections, from100% to 1%. Two check box can also be included. The Add check box, oncechecked, adds the presbyopia to the simulated eye. The Test check box,when checked, performs the distance (zero vergence). At the bottom,there is a slider with which all the saved images (e.g. PSF andconvolved images) can be reviewed.

There are many factors that can affect the pupil size, and these factorscan be considered optimization approaches of the present invention. Forexample, the shape can be customized for various lighting andaccommodation conditions. As shown in FIG. 14, and further discussed inTable 2, pupil size can change with lighting conditions. Each of thepresbyopia-mitigating and/or treating methods, devices, and systemsdescribed herein may take advantage of these variations in pupil size. Apupil size of a particular patient will often be measured, and multiplepupil sizes under different viewing conditions may be input for thesetechniques.

TABLE 2 dim bright distance 5 mm 3.5 mm near 4 mm 2.5 mm

A patient can also have a task-related vision preference that correlateswith lighting conditions, such as those described in Table 3, and thecustomization can be based upon these task-related preferences.

TABLE 3 cd/m² lighting condition 30 subdued indoor lighting 60display-only workplaces 120 typical office 240 bright indoor office 480very bright; precision indoor tasks 960 usual indoors 1920 brightafternoon

FIG. 15 illustrates that pupil size can change with accommodation, andFIG. 16 illustrates a comparison of corrections by providing optimizervalues for various accommodations. With 3 or more diopters of residualaccommodation, the optimizer value can achieve a limit of about 1.0,regardless of the pupil size. Typically, a larger amount of residualaccommodation can correspond to a smaller optimizer value afteroptimization. The limit line can correspond to an optimizer value ofabout 5.0. In other words, an optimizer value of about 5.0 can be viewedas a good practical limit. Either there can be a smaller pupil, or alarger amount of residual accommodation, in order to optimize such thatall vergence distances have good visual performance.

FIGS. 17 and 18 show optimizations under various accommodationconditions. FIGS. 18A and 18B show CMTF and optimizer values when pupilsize changes and Residual Accommodation (RA) are modeled. FIG. 18C showssimulated eye charts seen at different target distances afteroptimization, all assuming a 5 mm maximum pupil size. Each eye chart has2/100, 20/80, 20/60, 20/40, and 20/20 lines. The top line simulates noaccommodation and no pupil size changes. The middle line assumes noaccommodation but the pupil size changes from 5 mm (dim distance) to 2.5mm (bright near). In the bottom line, the simulation assumes 1 Daccommodation with pupil size changes from 5 mm (dim distance) to 2.5 mm(bright near).

FIG. 19 shows CMTF values for various corrections. A 5 mm pupil eye isassumed, along with a smallest pupil size of 2.5 mm (bright lightreading condition) and a 1 D residual accommodation. FIG. 20 comparesbi-focal, optimal, and multi-focal corrections, under the assumption ofa one diopter residual accommodation. These simulated eye charts areseen at different target distances after optimization. 1 D accommodationand a 5 mm pupil changes from 5 mm (dim distance) to 2.5 mm (brightnear) are assumed. The eye chart has 2/100, 20/80, 20/60, 20/40, and20/20 lines, respectively. FIG. 21 illustrates a simulated eye chartseen at different target distances. The data in this figure based on theassumption that the pupil size decreases from 5 mm to 2.5 mm, and thereis a 1 diopter residual accommodation in all cases.

The customized shape methods and systems of the present invention can beused in conjunction with other optical treatment approaches. Forexample, co-pending U.S. provisional patent application No. 60/431,634,filed Dec. 6, 2002 and co-pending U.S. provisional patent applicationNo. 60/468,387 filed May 5, 2003, the disclosures of which are herebyincorporated by reference for all purposes, describe an approach todefining a prescription shape for treating a vision condition in aparticular patient. The approach involves determining a prescriptiverefractive shape configured to treat the vision condition, theprescriptive shape including an inner or central “add” region and anouter region. The approach also includes determining a pupil diameter ofthe particular patient, and defining a prescription shape comprising acentral portion, the central portion having a dimension based on thepupil diameter, the inner region of the prescriptive refractive shape,and an attribute of at least one eye previously treated with theprescriptive refractive shape.

Accordingly, the present invention can include a method for determininga customized shape that includes a scaled central portion as describedabove, the customized shape giving results at least as good or betterthan previously known methods.

Systems

The present invention also provides systems for providing practicalcustomized or optimized prescription shapes that mitigate or treatpresbyopia and other vision conditions in particular patients. Thesystems can be configured in accordance with any of the above describedmethods and principles.

For example, as shown in FIG. 22, a system 100 can be used forreprofiling a surface of a cornea of an eye 150 of a particular patientfrom a first shape to a second shape having correctively improvedoptical properties. System 100 can comprise an input 110 that accepts aset of patient parameters, a module 120 that determines an opticalsurface shape for the particular patient based on the set of patientparameters, using a goal function appropriate for presbyopia of an eye,a processor 130 that generates an ablation profile, and a laser system140 that directs laser energy onto the cornea according to the ablationprofile so as to reprofile a surface of the cornea from the first shapeto the second shape, wherein the second shape corresponds to theprescription shape.

Defining a Scaled Prescription Shape for a Vision Condition

Determining a Prescriptive Prescription Shape

Certain prescriptive refractive shapes are effective in treating visionconditions, and it is possible to provide an efficient prescriptionshape by scaling a shape to the particular patient being treated.Optical shapes can be scaled based on data collected from subjectspreviously treated with a uniform prescriptive optical shape, such asmeasured manifest powers for different pupil sizes. Shapes may also bescaled based on the desired overall optical power of the eye underdiffering viewing conditions.

It is useful to select or construct an initial prescriptive refractiveshape appropriate for the vision condition. For example, prescriptivetreatment shapes such as those shown in FIG. 23 have been found toprovide a range of good focus to the eye so as to mitigate presbyopia.This particular prescriptive shape is the sum of two component shapes: abase curve treatment defining an outer region having a diameter of about6.0 mm, and a refractive add defining an inner region having a diameterof about 2.5 mm. Prescriptive shapes such as this can provide aspherical power add ranging from between about 1.0 diopters to about 4.0diopters at the inner region. Further, the spherical power add can beabout 3.1 diopters. Combining the inner and outer regions, the overallprescriptive refractive shape can be aspheric. It is appreciated,however, that the dimensions and properties of a prescriptive shape canvary depending on the intended purpose of the shape.

Treatment of presbyopia often involves broadening the focus range of theeye. Referring to FIG. 24, in an emmetropic eye a focal length of theoptical system results in a point of focus 10 that produces a sharpimage. At this point, the refractive power of the cornea and lens ismatched to the length of the eye. Consequently, light rays 20 enteringthe eye converge on the retina 30. If there is a difference between therefractive power and the length of the eye, however, the light rays canconverge at a point 40 in front of or behind the retina, and the imageformed on the retina can be out of focus. If this discrepancy is smallenough to be unnoticed, it is still within the focus range 50 or depthof focus. In other words, the image can be focused within a certainrange either in front of or in back of the retina, yet still beperceived as clear and sharp.

As shown in FIG. 25, when an object is at a far distance 60 from theeye, the light rays 20 converge on the retina 30, at focal point 10.When the object is moved to a near distance 70, the light rays 20′converge at a focal point 80 beyond the retina. Because the image isoutside of the depth of focus 50, the image is perceived to be blurred.Through the process of accommodation, the lens changes shape to increasethe power of the eye. The power increase brings the focal point 80 backtoward the retina as the eye attempts to reduce the blur.

In the presbyopic eye the accommodative mechanism may not worksufficiently, and the eye may not be able to bring the focal point tothe retina 30 or even within the range of focus 50. In thesecircumstances, it is desirable to have an optical system having abroadened focus range 50′. One way to achieve this is by providing anoptical system with an aspheric shape. The aspheric shape, for example,can be ablated on a surface of the eye, the surface often comprising astromal surface formed or exposed by displacing or removing at least aportion of a corneal epithelium, or a flap comprising cornealepithelium, Bowman's membrane, and stroma. Relatedly, the shape can beprovided by a correcting lens. In some optical systems, only a portionof the shape may be aspheric. With an aspheric shape, there is not asingle excellent point of focus. Instead, there is greater range of goodfocus. The single best focus acuity is compromised, in order to extendthe range of focus. By extending the range of focus 50 to a broadenedrange of focus 50′, there is an improvement in the ability to see bothdistant and near objects without the need of 3 D or more in residualaccommodation.

Without being bound by any particular theory, it is believed that thepower add of the inner region depicted in FIG. 23 provides a myopiceffect to aid near vision by bringing the near vision focus closer tothe retina, while the outer region remains unaltered for distancevision. In this sense the application of this prescriptive shape isbifocal, with the inner region being myopic relative to the outerregion. Put another way, the eye can use the inner region for nearvision, and can use the whole region for distance vision.

In a laser ablation treatment, the prescriptive refractive ablationshape can have fairly abrupt changes, but post ablation topographies mayshow that healing of the eye can smooth the transitions. The shape canbe applied in addition to any additional required refractive correctionby superimposing the shape on a refractive corrective ablation shape.Examples of such procedures are discussed in co-pending U.S. patentapplication Ser. No. 09/805,737, filed Mar. 13, 2001 the disclosure ofwhich is herein incorporated by reference for all purposes.

Alternative presbyopia shapes may also be scaled using the techniquesdescribed herein, optionally in combination with other patientcustomization modifications, as can be understood with reference to U.S.Provisional Patent Application Nos. 60/468,387 filed May 5, 2003,60/431,634, filed Dec. 6, 2002, and 60/468,303, filed May 5, 2003, thedisclosures of which are herein incorporated by reference for allpurposes. Alternative presbyopia shapes may include concentric addpowers along a peripheral or outer portion of the pupil, along anintermediate region between inner and outer regions, along intermittentangular bands, or the like; asymmetric (often upper or lower) addregions, concentric or asymmetric subtrace or aspheric regions, and thelike. The present application also provides additional customizedrefractive shapes that may be used to treat presbyopia.

Determining a Pupil Diameter of the Particular Patient

When scaling a refractive shape to treat a particular patient, it ishelpful to determine the pupil diameter of the particular patient to betreated. Several methods may be used to measure the pupil diameter,including image analysis techniques and wavefront measurements such asWavescan® (VISX, Incorporated, Santa Clara, Calif.) wavefrontmeasurements. The size of the pupil can play a role in determining theamount of light that enters the eye, and can also have an effect on thequality of the light entering the eye. When the pupil is veryconstricted, a relatively small percentage of the total light falling onthe cornea may actually be allowed into the eye. In contrast, when thepupil is more dilated, the light allowed into the eye may correspond toa greater area of the cornea. Relatedly, the central portions of thecornea have a more dominant effect on the light entering the eye than dothe peripheral portions of the cornea.

Pupil size can have an effect on light quality entering the eye. Whenthe pupil size is smaller, the amount of light passing through thecentral portion of the cornea is a higher percentage of the total lightentering the eye. When the pupil size is larger, however, the amount oflight passing through the central portion of the cornea is a lowerpercentage of the total light entering the eye. Because the centralportion of the cornea and the peripheral portion of the cornea candiffer in their refractive properties, the quality of the refractedlight entering a small pupil can differ from that entering a largepupil. As will be further discussed below, eyes with different pupilsizes may require differently scaled refractive treatment shapes.

An Inner Region of the Prescriptive Refractive Shape

Experimental data from previously treated eyes can provide usefulinformation for scaling a refractive treatment shape for a particularpatient. For example, a refractive shape for a particular patient can bescaled based on certain characteristics or dimensions of the shape usedto treat the eyes of the subjects. One useful dimension of theabove-described presbyopic prescriptive shape is a size or diameter ofinner region or refractive add. It is possible to scale a treatmentshape for a particular patient based on the diameter of the refractiveadd of the prescriptive shape. Alternative techniques might scale apower of an inner, outer, or intermediate region, a size of an outer orintermediate region, or the like.

If the refractive add diameter is small, it can occupy a smallerpercentage of the total refractive shape over the pupil. Conversely, ifthe refractive add diameter is large, it can occupy a greater percentageof the total refractive shape over the pupil. In the latter case,because the area of the periphery is relatively smaller, the distancepower is diminished. In other words, the area of the add is taking upmore of the total refractive shaped used for distance vision.

An Attribute of a Set of Eyes Previously Treated with the PrescriptiveRefractive Shape

As noted above, experimental data from previous prescriptive eyetreatments can be useful in scaling a treatment for a particularindividual. When scaling a presbyopia treatment shape, it is helpful toidentify a pupil diameter measure from among a set of previously treatedeyes having a fixed treatment size that corresponds to both gooddistance and near sight. It is possible to use acuity and powermeasurements from the set of treated eyes to determine such a pupildiameter. The fixed treatment size (e.g. 2.5 mm inner region) can thenbe said to be appropriate for this identified pupil diameter.

FIGS. 26 and 27 illustrate the effect that pupil size can have ondistance acuity and near acuity in subjects treated with a prescriptiverefractive shape, for example a shape having a 2.5 mm central add zoneof −2.3 diopters. Referring to FIG. 26, pupil size values were obtainedfrom a group of subjects as they gazed into infinity under mesopic ordim light conditions. The 6-month uncorrected distance acuity valueswere obtained from the same group of subjects under photopic conditions.Referring to FIG. 27, pupil size values were obtained from a group ofsubjects as they gazed at a near object under mesopic or dim lightconditions. The 6-month uncorrected near acuity values were obtainedfrom the same group of subjects under photopic conditions.

One way to determine an optimal pupil diameter measure is bysuperimposing a near acuity graph over a distance acuity graph, andascertaining the pupil diameter that corresponds to the intersection ofthe lines.

Another way to determine a pupil diameter that corresponds to both gooddistance and near acuity is to define each of the slopes mathematically:Near acuity=−2.103+0.37879*Pupil size(Dim)  (FIG. 27)Distance acuity=0.40001−0.0677*Pupil size(Dim)  (FIG. 26)By setting the two equations from the graphs equal, it is possible tosolve for the intersection point.−2.103+0.37879*Pupil size(Dim)=0.40001−0.0677*Pupil size(Dim)Pupil size(Dim)=2.4/0.45=5.33 mm

An optimum overlap can occur in a range from between about 4.0 mm toabout 6.0 mm. Further, an optimum overlap can occur in a range frombetween about 5.0 mm to about 5.7 mm. These measurements may correspondto a pupil diameter measure from the set of previously treated eyes thatcorresponds to both good distance and near vision when the diameter ofthe central add region is 2.5 mm.

Defining a Refractive Shape for Treating a Particular Patient Acuity asa Function of Pupil Size

The present invention provides methods and systems for defining aprescription for treating a vision condition in a particular patient,with the prescription optionally comprising a refractive shape. Such amethod can be based on the following features: (a) a prescriptiverefractive shape configured to treat the vision condition, including aninner region thereof; (b) a pupil diameter of the particular patient,and (c) an attribute of a set of eyes previously treated with theprescriptive shape.

For example, the prescriptive shape can be the shape described in FIG.23. The inner region of the shape can be a refractive add, having adiameter of 2.5 mm. For illustrative purposes, a pupil diameter of theparticular patient of 7 mm is assumed. The attribute of a set ofpreviously treated eyes can be the pupil diameter of the eyes thatcorresponds to both good distance and near vision, such as the exemplary5.3 mm treated pupil diameter shown in FIGS. 26 and 27. Thus, a ratio ofthe prescriptive refractive add to treated pupil (PAR) can be expressedas 2.5/5.3.

The PAR can be used in conjunction with the pupil diameter of theparticular patient to scale the refractive shape. For example, a centralportion of the scaled refractive shape can be calculated as follows.central portion diameter=PAR*pupil diameter of particular patientGiven the example above, the diameter of a central portion of the scaledrefractive shape for treating the particular patient is:(2.5/5.3)*7 mm=3.3 mm

In this example, this scaled central portion can correspond to thediameter of the refractive add of the defined refractive shape. Itshould be appreciated that the refractive shape and the central portionof the refractive shape can alternately be spheric or aspheric. Forexample, the refractive shape can be aspherical, and the central portionof the refractive shape can be aspherical; the refractive shape can bespherical and the central portion of the refractive shape can bespherical; the refractive shape can be aspherical, and the centralportion of the refractive shape can be spherical; or the refractiveshape can be spherical, and the central portion of the refractive shapecan be aspherical.

As shown above, the PAR can be about 2.5/5.3, or 0.47. It will beappreciated that the PAR can vary. For example, the PAR can range frombetween about 0.35 and 0.55. In some embodiments, the PAR may range fromabout 0.2 to about 0.8. Optionally, the PAR can range from about 0.4 toabout 0.5. Further, the PAR can range from about 0.43 to about 0.46. Itwill also be appreciated that the ratios discussed herein can be basedon area ratios or on diameter ratios. It should be assumed that whendiameter ratios are discussed, that discussion also contemplates arearatios.

Power as a Function of Pupil Size

In another example, the attribute of a set of previously treated eyescan be the pupil diameter of the eyes that correspond to both gooddistance and near values for spherical manifest. A group of individualswith varying pupil sizes were treated with the same prescriptiverefractive shape, the shape having a constant presbyopic refractive adddiameter of approximately 2.5 mm. Pupil sizes were obtained on aWavescan® device. The Spherical Manifest at 6 months post-treatment isshown as a function of the pupil size in FIG. 28. Here, the sphericalmanifest represents the effective distance power as the result from thetotal prescriptive shape, including the inner region and outer regionsof the shape.

As FIG. 28 illustrates, for a given prescriptive treatment shape, theeffect that the shape has on the individual's manifest can depend on theindividual's pupil diameter. Depending on the pupil size of the treatedsubject, the refractive add will have different relative contribution tothe power. And due to the varying pupil sizes, the prescriptiverefractive add to treated pupil ratio (PAR) may not be constant. Thus,with the same prescriptive treatment, the effective power can vary amongdifferent patients. In a simplified model, the power change from thecentral portion of the treated eye to the periphery can be assumed to belinear. This simplification can be justified by the data. The change inpower can be represented by the following formula, expressed in units ofdiopters.MRS(Effective Distance Power)=−2.87+0.42*Pupil size (Dim)[diopters]

The rate change in effective power is 0.42 D per mm for distance vision.It has been shown that the pupil diameter can change at a rate ofapproximately 0.45 D per mm. The add power is −2.87 diopters.

Without being bound by any particular theory, it is thought that due tothe asphericity of the central add, there can be a linear relationshipbetween the effective distance power and the pupil diameter.Accordingly, is it possible to characterize the ratio of effectivedistance power versus pupil diameter with the following linear coreequation, where C₀ and A are constants.Effective Distance Power=C₀ +A(pupil_diameter)  Equation A

In individuals having smaller pupil diameters, the contribution of theouter region of the prescriptive shape is diminished; the manifestrefraction is more myopic and the effective power is smaller. Andwhereas a lower MRS value can correspond to a more myopic refraction, ahigher MRS value can correspond to a less myopic refraction. Themanifest refraction, which can be expressed in terms of power, is oftenproportional to distance vision, which can be expressed in terms ofacuity or logarithm of the minimum of angle of resolution (logMAR).

As discussed above, a PAR can be determined based on acuity measurementsas a function of pupil size. In an analogous manner, it is possible todetermine a PAR based on power measurements as a function of pupil size.

Skewing

The Effective Distance Power Equation A above represents one approach tofinding a good approximation to customize the refractive shape size. Insum, the intersection of a distance version of the equation and a nearversion of the equation is solved to determine a pupil diameter measure,which forms the denominator for the PAR (prescriptive shape adddiameter/pupil diameter of treated eye). By adjusting the PAR, it ispossible to adjust the shape to achieve emmetropia or other refractivestates.

Altering the Size of the Prescriptive Shape Add

Referring to FIG. 28, a treated pupil diameter of about 5.4 mm has aspherical manifest of about −0.6 diopters. If the size of theprescriptive shape add is made bigger, the line can be shifted downward.Consequently, the effect in a particular patient treated with the scaledrefractive shape would be a more myopic spherical manifest of −2.0, forexample. On the other hand, if the size of the add is made smaller, theline can be shifted upward, and the effect would be a spherical manifestof −0.2, for example. As the diameter of the add decreases, the manifestof the particular patient treated with the scaled refractive shapebecomes more skewed to better distance sight. As the diameter of the addincreases, the manifest becomes more skewed to better near sight.

Fixing the PAR

It is possible to set the near manifest for all patients by fixing thePAR. Referring to the example of FIGS. 26 and 27 (where the Equation Aintersection is about 5.3 mm), a ratio of 2.5/5.3 mm can rotate thesenear and distance lines toward horizontal, about the 5.3 mm point. Inother words, an analysis of particular patients treated with a PAR of2.5/5.3 is expected to result in manifest versus pupil size plots havinglines that are more horizontally oriented. Thus each patient would beexpected to have similar near manifest. Alternatively, it is possible tochoose a different point of rotation to optimize distance manifest overnear manifest, or vice versa. For example, by choosing a 5.0 mm pointfor rotation, better near manifest can be provided at the expense of thedistance manifest.

When comparing the graphs of FIGS. 26 and 27 the distance acuity andnear acuity slopes can vary. As shown in these figures, near visionchanges at a slightly higher rate than distance vision. In other words,near vision appears to be more sensitive to changes in pupil diameterthan distance vision. An adjustment was made to near measurements inFIG. 27 to offset a distance correction used during the measurement.

Non-Linear Models

The effective distance power versus pupil diameter can also be expressedby the following non-linear equation.Power=C ₀ +A(pupil_diameter)+B(pupil_diameter)² +C(pupil_diameter)³+ . ..   Equation Bwhere C₀, A, B, and C are constants. This equation is only one of manythat can be used to model the desired relationship. Similar non-linearequations can be used to model desired effective power, as discussedbelow. Also, both linear and non-linear equations can be used to modeltarget manifest, as discussed below.

Target Manifest (Acuity as a Function of Power)

The target manifest or desired power at a particular viewing distancemay or may not be emmetropic (0 diopters). For example, near sight maybe improved by a manifest which is slightly myopic. Following ananalysis similar to that discussed above for pupil size dependency, anoptimum target refraction can be calculated based on acuity as afunction of power in a set of eyes treated with the prescribedrefractive shape. FIGS. 29 and 30 show the distance and near acuity as afunction of manifest, respectively. Distance and near acuity versusmanifest can be expressed by the following non-linear equations.Near_Acuity=A ₀ +A(Manifest)+B(Manifest)² +C(Manifest)³+ . . .Distance_Acuity=A ₀ +A(Manifest)+B(Manifest)² +C(Manifest)³+ . . .

Applying a first order approximation to the above equations, and usingmeasurements from previous data, the near and distance acuity as afunction of manifest can be expressed as follows.Near_Acuity=0.34+0.67(Manifest)Dist_Acuity=−0.04−0.13(Manifest)

The intersection between the two functions can be solved as follows.

0.34 + 0.67  (Manifest) = −0.04 − 0.13  (Manifest) $\begin{matrix}{{Manifest} = \frac{\left( {{- 0.04} - 0.34} \right)}{0.67 + 0.13}} \\{= {- {0.48\mspace{14mu}\lbrack{Diopters}\rbrack}}}\end{matrix}$

The point where the two lines meet is about −0.5 D. Therefore, it can beuseful to set the target manifest to −0.5 D. The target manifestequations can be refined based on additional data collected from thosepatients that are treated with the refractive shape. As noted above inreference to FIG. 23, a prescriptive shape may be the sum of a basecurve treatment and a central refractive add. It is possible to changethe base shape to compensate for any power offset contributed by thecentral refractive add to the distance manifest.

PAR Refinements Applied to Particular Patients

As additional data is accumulated, it is possible to calculate thehigher order terms of Equation B. More particularly, it is possible tocalculate the higher order terms from additional subjects who have beentreated with refractive shapes corresponding to constant and linear termadjustments. For example, a group of patients can be treated accordingto the PAR of 2.5/5.3 discussed above, and based on their results, thePAR can be further refined.

A group of patients had adjustments made to their prescriptivepresbyopic shape based on results from the analysis discussed above. Thepatients were treated with shapes based on a constant PAR of 2.5/5.6 asapplied to the central add shape, with a target manifest of −0.5 D.These adjustments rotate the equation about the 5.6 mm line towardhorizontal because the near effect is a constant. For example, a 5 mmpupil patient has the same near correction as a 6 mm pupil patient,which means that their near acuity should be the same, i.e. a plot ofthe near acuity versus pupil size will be a substantially flat line.FIGS. 31 and 32 show the result of these adjustment on this group ofpatients. As predicted, the lines rotated. The distance acuity of 7 of 8of these patients was 20/20 (logMAR 0) or better, and the 8th was20/20+2. Their near acuity slopes have also flattened, with 7/8 patienthaving simultaneous 20/32−2 acuity or better, and the 8th 20/40. Table 6summarizes the acuity and power measures.

TABLE 4 Near acuity 0.19 ± 0.1 Distance acuity −0.08 ± 0.08 MRS −0.19 ±0.26

This PAR adjusted group has, which is a good result for a presbyopiatreatment.

Optimizing a Refractive Shape for a Vision Condition

It is possible to define customized refractive shapes such that they areoptimized to treat a particular patient. In one approach to defining anoptimized refractive shape, the power of the refractive shape may bebased on the central power add of a prescriptive shape, and the powerchange requirement of the particular patient. Other approaches mayinvolve deriving an appropriate prescription so as to provide a desiredoverall effective power of the eye at different viewing conditions,again by taking advantage of the changes in pupil size.

Determining a Desired Central Power Add of a Prescriptive RefractiveShape Configured to Treat the Vision Condition

A prescriptive shape can be selected for treating the vision conditionof the particular patient. For example, the prescriptive shape shown inFIG. 23 can be selected for treating a particular patient havingpresbyopia. As previously discussed, the central power add of thisexemplary prescriptive shape can be about −3.1 diopters.

Determining a Power Change of a Particular Patient

The desired power change of a particular patient can vary widely, andoften depends on the patient's desired treatment or a recommendationfrom a vision specialist. For example, the desired power change of aparticular patient having presbyopia can be about −2.5 diopters. Thedesired power change may be linear or non-linear.

Determining a Pupil Diameter Parameter of the Particular Patient

When defining a refractive shape for treating a vision condition in aparticular patient, it is helpful to determine the pupil diameterparameter of the particular patient. Pupil diameters can be measured by,for example, a pupillometer. Pupil diameter parameters can involve, forexample, the patient's pupil diameter as measured under certain distanceand lighting conditions, such as under photopic conditions while thepatient gazes at infinity (distance-photopic). Pupil diameter parameterscan also involve pupil diameter measurements under other conditions suchas distance-mesopic, distance-scotopic, near-photopic, near-mesopic, ornear-scotopic. Still further additional measurements at other viewingconditions, such as at intermediate distances and/or moderate lightingconditions, may also be measured. Often, pupil diameter parameters willbe based on two pupil diameter measurements. For example, a pupildiameter parameter can be the value of the particular patient's pupildiameter at distance-photopic minus the patient's pupil diameter atdistance scotopic. According to this example, if the distance-photopicpupil diameter is 0.7 mm and the distance-scotopic pupil diameter is 0.2mm, then the pupil diameter parameter is 0.7 mm minus 0.2 mm, or 0.5 mm.

Defining a Refractive Shape Configured to Treat the Particular Patient,the Power of the Refractive Shape at a Given Diameter Based on: theCentral Power Add of the Prescriptive Refractive Shape, the Power ChangeRequirement of the Particular Patient, and the Pupil Diameter Parameterof the Particular Patient

When defining the refractive treatment shape, it can be beneficial tobase the power of the refractive shape (Power/Shape Requirement) at agiven diameter based on the central power add of the prescriptiverefractive shape, and on the power change requirement of the particularpatient. For example, the power of the refractive shape can be afunction of a given diameter, as expressed in the following formula.Power/Shape_Requirement=C₀ +A(pupil_diameter)where Power/Shape Requirement is the power of the refractive shape at aparticular Pupil_Diameter, C₀ is the central power add of theprescriptive refractive shape, and A is calculated asA=(PRC−C ₀)/PDPwhere PRC is the power change requirement for the particular patient,and PDP is the pupil diameter parameter (obtained, for example, bysubtracting the diameter of the pupil measured when the patient isgazing at infinity from the diameter of the pupil measured when thepatient is looking at a near object under identical light conditions).Given the values discussed above, the Power/Shape_Requirement (PSR) canbe calculated as follows.PSR=−3.1 diopters+[(−2.5 diopters−−3.1 diopters)/0.5mm)](pupil_diameter)orPSR=−3.1 diopters+1.2(pupil_diameter)

Other Pupil Diameter Parameters

It is also possible to calculate a pupil diameter parameter based on apupil diameter change slope as measured under certain distance andlighting conditions, for example, as the patient gazes at infinity whilethe lighting conditions change from photopic to scotopic(distance-photopic to scotopic). Pupil diameter parameters can alsoinvolve pupil diameter change slopes such as near-photopic to scotopic,photopic-distance to near, mesopic-distance to near, orscotopic-distance to near.

The Effective Power

The effective power (e.g., linear power model or higher order model) canbe used to calculate or derive a presbyopic shape, optionally based onthe following parameters.

-   -   F.1. Emmetropic at distance (photopic and mesopic lighting        conditions)        -   a. This can determine a maximum diameter of the add    -   F.2. Near can have an effective power of −2.5 D (or more, if        desired by the patient    -   F.3. The rate of change of power for the add-treatment        combination can have one of the four:        -   i. The same power rate of change as the photopic—Distance to            near        -   ii. The same power rate of change as the mesopic—Distance to            near        -   iii. The same power rate of change as the scotopic—Distance            to near        -   iv. Non-linear rate of change similar to the above, but is            optimized to give better simultaneous distance and near            vision.

For an eye gazing into infinity, under photopic conditions, thetheoretical pupil size at emmetropia can vary within the population.Moreover, the pupil diameter can further vary when the eye is used fordifferent tasks. For example, the pupil diameter can decrease as theeye's gaze changes from infinity to a near object. As the eye changesfrom a distance gaze to a near gaze, the typical pupil diameterdecreases. This change in pupil diameter may be linear with convergenceand sigmoid with accommodation. In an eye treated with an exemplaryprescriptive shape, the pupil diameter at near gaze can typically havethe inner region of the prescriptive shape as the dominant refractivecomponent. Consequently, the change of pupil size from larger to smaller(distance gaze to near gaze) can be equivalent to a change in power. Incomparison, the distance gaze pupil will have an effective power basedon the combination of the inner region add and the outer region of theprescriptive shape, with the outer region becoming a more dominantrefractive component. Therefore, each refractive shape can be customizedto each particular individual because of the many different combinationsavailable. By changing the power of the cornea, for example, fromemmetropia at the “distance” pupil size to within a range of about −1.0diopters to about −4.0 diopters myopic for “near” pupil size, it may bepossible to mitigate presbyopia.

A general prescription may go as follows. First, measure the continuouspupil size and/or size change at different distances and lightingconditions, such as for at least one (optionally two or more, in somecases all) of: Distance-Photopic; Distance-Mesopic, Distance-Scotopic,Near-Photopic, Near-Mesopic, and/or Near-Scotopic. The pupil size can beaffected by the lighting conditions as well as viewing distances. Therefractive shape can also include adjustments and/or optimization forlighting. In photopic conditions, the pupil is typically constricted. Inscotopic conditions, the pupil is usually dilated. Under mesopicconditions, the pupil can be variably dilated or constricted dependingon the specific type of mesopic condition. Second, calculate the pupildiameter continuous rate of change for the following combinations:Distance-photopic to scotopic, Near-photopic to scotopic,Photopic-Distance to near, Mesopic-Distance to near, and/orScotopic-Distance to near. It is possible to design a shape and ablationsize such that patient is substantially emmetropic as pupil size goesfrom larger (distant) to smaller (near), typically within a range. Thepresbyopic lens power can compensate focus such that the lens is theinverse of the rate of pupil change. To do this, the power can change(for example—3 D) for different pupil diameters.

Power/Shape_Requirement = C₀ + A(pupil_diameter) + B(pupil_diameter)² + C(pupil_diameter)³ + …

The Power/Shape_Requirement in the above equation may be effectivepower, and/or may be manifest power. The power can change with changesin pupil diameter. For a linear power shape, the coefficient A can becalculated as follows.

$\frac{d({power})}{d({pupil\_ diameter})} = A$Solving for the linear coefficient,

$A = \frac{{PowerChangeRequirement} - C_{0}}{{pupil\_ diameter}{\_ rate}{\_ of}{\_ change}}$

The target manifest can be targeted to the patient's request or adoctor's recommendation by using the effective distance power equationas described above in the “target manifest” section.

Multifocal Shapes

A good refractive shape (including a multi-focal shape) may be at ornear an optimum compromise between distance and near sight. The near addhas an “effective” power—it may not have a single power because of themulti-focal shape. The sum of the peripheral and central add may givethe distance power—again it may not have a single power because of themulti-focal shape.

The Age Dependent Presbyopic Shape

As discussed above, as one ages, accommodation decreases. This is shownin FIG. 33. At 60, accommodation can decrease significantly, even tonearly zero. Studies have shown that pupil sizes decrease as one getsolder. As seen in the figure, the slope or rate of change inaccommodation also changes with age. It is possible to optimize thepupil dependencies to the age related change in accommodation. The rateof distance and near acuities for a central add shape can beNear_acuity=−2.103+0.37879*Pupil size(Dim)Distance acuity=0.40001−0.0677*Pupil size(Dim)

According to these equations, as the pupil size decreases, the nearacuity gets better, at a rate of 0.37 lines per millimeter. The distanceacuity gets worse, but at much slower rate of 0.07 lines per millimeter.Therefore, it is possible to optimize the treatment parameters for thepatient's age by targeting the treatment for less myopia. It is possibleto allow a shift in the centering of the “range” by taking the residualaccommodation into account in the customization of the treatment.

It is possible that the optimum shape may be on a “linear” powerapproximation as discussed above, but it may consist of higher orders.Though the effective power can be given by the equation above, the shapecan be constant over, for example, a central 2.5 mm and have a curvaturegradient that will blend the central add to the peripheral region. Withthis shape it may be beneficial to choose the diameter of the centraladd to match the patients near pupil such that the near pupil willencompass only the central add when it's at its smallest, and thegradient will be customized to the patient's pupil size rate of change.

Hence, by modeling the residual accommodation, the range of pupil changemay be shifted to optimize the “life” long presbyopic correction.

Systems

The present invention also provides systems for scaling refractiveshapes and providing practical customized or optimized refractive shapesthat mitigate or treat presbyopia and other vision conditions inparticular patients. The systems can be configured in accordance withany of the above described methods and principles.

For example, as shown in FIG. 34, a system 1000 can be used forreprofiling a surface of a cornea of an eye 1600 of a particular patientfrom a first shape to a second shape having correctively improvedoptical properties. System 1000 can comprise an input 1100 that acceptsa prescriptive shape specific for treating the vision condition, aninput 1200 that accepts a pupil dimension of the particular patient, amodule 1300 that scales a dimension of a central portion of a refractiveshape based on the pupil dimension of the particular patient and anattribute of at least one eye previously treated with the prescriptiveshape, a processor 1400 that generates an ablation profile, and a lasersystem 1500 that directs laser energy onto the cornea according to theablation profile so as to reprofile a surface of the cornea from thefirst shape to the second shape, wherein the second shape corresponds tothe refractive shape.

Calculating of Presbyopia Mitigating Prescriptions

Methods, Systems, and Devices described herein can be used to generateprescriptions for treatment of refractive errors, particularly fortreatment of presbyopia. Such treatments may involve mitigation ofpresbyopia alone, or may treat a combination of presbyopia with otherrefractive disorders.

As described above, presbyopia is a condition where the degree ofaccommodation decreases with the increase of age. Most people have somedegree of presbyopia by the age of about 45.

Treatments of presbyopia may involve passive and/or active procedures.In passive procedures, treatment or mitigation is performed in such away that an improved balance between near vision and distance vision isprovided and maintained. In an active procedure, restoration of full orpartial accommodation is a goal. So far, active procedures for thecorrection of presbyopia have not been fully successful.

With passive procedures, it is desirable to provide an improved and/oroptimal balance between near vision and distance vision. In order to dothat, patients may sacrifice some of their distance vision to gainimproved near vision. In addition, they may sacrifice some contrastsensitivity because of the introduction of the asphericity of the newoptics of the eye. Fortunately, the sacrifice of distance vision andcontrast sensitivity may be mitigated by taking advantage of a pupilshrinkage when the eye accommodates.

As described below, an analytical solution for a presbyopia shape can beachieved based on a desire for different powers at different pupilsizes. In order to understand this, we can take advantage of a conceptof optical power that depends on the change of pupil size and might alsodepend on wavefront aberrations other than defocus terms. We willconcentrate on the pupil size dependency in this description.

The following approach considers the correction as a “full pupil”correction rather than “partial pupil” correction as employed with acentral add. Healing effect, flap effect as well as how the effectivepower correlates with the manifest refraction may be addressed withempirical studies, allowing these effects to be fed back into thefollowing calculations and/or a laser ablation planning program asappropriate so as to provide optimized real-world results.

Effective Power and its Application to Presbyopia

As used herein, “effective power” means the optical power that bestmatches the manifest sphere at a certain pupil size. With wavefrontbased ocular aberrations, the defocus-dependent effective power can bewritten as

$\begin{matrix}{{P_{eff} = {- \frac{4\sqrt{3}c_{2}^{0}}{R^{2}}}},} & (1)\end{matrix}$where R stands for the pupil radius in mm when c₂ ⁰ is the Zernikecoefficient given in microns in order to get the effective power indiopters, and P_(eff) is effective power. When a wavefront map isdefined in radius R with a set of Zernike polynomials, when the pupilshrinks the smaller map, if re-defined with a new set of Zernikepolynomials, will have a different set of Zernike coefficients than theoriginal set. Fortunately, analytical as well as algorithmaticalsolutions of the new set of Zernike coefficients exist. If the originalset of Zernike coefficients is represented by {c₁} that corresponds topupil radius r₁, then the new set of Zernike coefficients {b_(i)} thatcorresponds to pupil radius r₂ can be expressed by a recursive formulaas

$b_{2i}^{0} = {{e^{2i}{\sum\limits_{j = 0}^{{n/2} - i}{\left( {- 1} \right)^{j}c_{2{({i + j})}}^{0}\sqrt{\frac{{2\left( {i + j} \right)} + 1}{{2i} + 1}}\frac{\left( {{2i} + j} \right)!}{{j!}{\left( {2i} \right)!}}}}} - {\sum\limits_{\underset{{step}\mspace{14mu} 2}{k = {2{({i + 1})}}}}^{n}{b_{k}^{0}\sqrt{\frac{k + 1}{{2i} + 1}}\frac{\left( {- 1} \right)^{{k/2} - i}{\left( {{k/2} + i} \right)!}}{{\left( {{k/2} - i} \right)!}{\left( {2i} \right)!}}}}}$where e=r₂/r₁, n is the maximum radial order. As an example, if we seti=1, and n=4, we have the following formulab ₂ ⁰ =[c ₂ ⁰−√{square root over (15)}(1−e ²)c ₄ ⁰ ]e ²Therefore, a power profile with pupil size can be given as a conditionto obtain an optical surface for presbyopia correction.

In order to obtain a presbyopia prescription (which will here be anoptical shape), let's assume that we know the power profile or desiredeffective optical powers for different viewing conditions so as tomitigate presbyopia. From the power profile, we can in general do anintegration to calculate the wavefront shape. In the following, weconsider three cases where two, three, or four power points (differentdesired effective optical powers for different associated viewingconditions, often being different viewing distances and/or pupildiameters) are known.

Two-Power-Point Solution

Let's consider radially symmetric terms Z₂ ⁰ and Z₄ ⁰, when the pupilradius is changed from R to eR, where e is a scaling factor not largerthan 1, since the new set of Zernike coefficients for the defocus termcan be related to its original coefficients asb ₂ ⁰ =[c ₂ ⁰−√{square root over (15)}(1−e ²)c ₄ ⁰ ]e ².  (2)Substituting c₂ ⁰ with b₂ ⁰, and R² with e²R² in Equation 1 usingEquation 2, we have4√{square root over (3)}c ₂ ⁰−12√{square root over (5)}(1−e ²)c ₄ ⁰ =−R² P.  (3)

Suppose we request power p₀ at radius e₀R, and p₁ at radius e₁R, ananalytical solution of the original wavefront shape, which isrepresented by c₂ ⁰ and c₄ ⁰, can be obtained as

$\begin{matrix}{{c_{2}^{0} = {{- \frac{{\left( {1 - e_{1}^{2}} \right)p_{0}} - {\left( {1 - e_{0}^{2}} \right)p_{1}}}{4\sqrt{3}\left( {e_{0}^{2} - e_{1}^{2}} \right)}}R^{2}}}{c_{4}^{0} = {{- \frac{p_{0} - p_{1}}{12\sqrt{5}\left( {e_{0}^{2} - e_{1}^{2}} \right)}}{R^{2}.}}}} & (4)\end{matrix}$

As an example, let's consider a pupil with a dim distance size of 6 mm,requesting effective power of 0 D at pupil size 6 mm and bright readingpupil size of 4.5 mm, requesting effective power of −1.5 D. Substitutinge₀=6/6=1, e₁=4.5/6=0.75, and p₀=0 and p₁=−1.5, we get c₂ ⁰=0 and c₄⁰=−1.15. FIGS. 35 and 36 show the presbyopia shape and effective poweras a function of pupil size. It is very close to a linear relationship.

Three-Power-Point Solution

Let's consider radially symmetric terms Z₂ ⁰, Z₄ ⁰ and Z₆ ⁰, when thepupil radius is changed from R to eR, where e is a scaling factor notlarger than 1, since the new set of Zernike coefficients for the defocusterm can be related to its original coefficients asb ₂ ⁰ =[c ₂ ⁰−√{square root over (15)}(1−e ²)c ₄ ⁰+√{square root over(21)}(2−5e ²+3e ⁴)c ₆ ⁰ ]e ²,  (5)Substituting c₂ ⁰ with b₂ ⁰, and R² with e²R² in Equation 1 usingEquation 5, we have4√{square root over (3)}c ₂ ⁰−12√{square root over (5)}(1−e ²)c ₄⁰+12√{square root over (7)}(2−5e ²+3e ⁴)c ₆ ⁰ =−R ² P.  (6)Suppose we request power p₀ at radius e₀R, p₁ at radius e₁R, and p₂ andradius e₂R, an analytical solution of the original wavefront shape,which is represented by c₂ ⁰, c₄ ⁰ and c₆ ⁰, can be obtained as

$\begin{matrix}{{c_{2}^{0} = {{- \frac{\begin{matrix}\begin{matrix}{{\left( {1 - e_{1}^{2}} \right)\left( {1 - e_{2}^{2}} \right)\left( {e_{1}^{2} - e_{2}^{2}} \right)p_{0}} -} \\{{\left( {1 - e_{0}^{2}} \right)\left( {1 - e_{2}^{2}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)p_{1}} +}\end{matrix} \\{\left( {1 - e_{0}^{2}} \right)\left( {1 - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{1}^{2}} \right)p_{2}}\end{matrix}}{4\sqrt{3}\left( {e_{1}^{2} - e_{2}^{2}} \right)\left( {e_{0}^{2} - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)}}R^{2}}}{c_{4}^{0} = {{- \frac{\begin{matrix}\begin{matrix}{{\left( {5 - {3e_{1}^{2}} - {3e_{2}^{2}}} \right)\left( {e_{1}^{2} - e_{2}^{2}} \right)p_{0}} -} \\{{\left( {5 - {3e_{0}^{2}} - {3e_{2}^{2}}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)p_{1}} +}\end{matrix} \\{\left( {5 - {3e_{0}^{2}} - {3e_{1}^{2}}} \right)\left( {e_{0}^{2} - e_{1}^{2}} \right)p_{2}}\end{matrix}}{36\sqrt{5}\left( {e_{1}^{2} - e_{2}^{2}} \right)\left( {e_{0}^{2} - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)}}R^{2}}}{c_{6}^{0} = {{- \frac{{\left( {e_{1}^{2} - e_{2}^{2}} \right)p_{0}} - {\left( {e_{0}^{2} - e_{2}^{2}} \right)p_{1}} + {\left( {e_{0}^{2} - e_{1}^{2}} \right)p_{2}}}{36\sqrt{7}\left( {e_{1}^{2} - e_{2}^{2}} \right)\left( {e_{0}^{2} - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)}}R^{2}}}} & (7)\end{matrix}$

As an example, let's consider a pupil with WaveScan pupil size of 6 mm,and dim distance pupil size of 6 mm, requesting effective power of 0 Dand bright reading pupil of 3.5 mm, requesting effective power of −1.5D. In between are the dim reading and bright distance, with combinedpupil size of 4.5 mm with effective power of −0.5 D. Substitutinge₀=6/6=1, e₁=4.5/6=0.75, and e₂=3.5/6=0.583 as well as p₀=0, p₁=−0.6 andp₂=−1.5, we get c₂ ⁰=0, c₄ ⁰=−0.31814 and c₆ ⁰=0.38365. FIGS. 37 and 38shows the presbyopia shape and the effective power as a function ofpupil sizes.

Four-Power-Point Solution

Let's consider radially symmetric terms Z₂ ⁰, Z₄ ⁰, Z₆ ⁰ and Z₈ ⁰, whenthe pupil radius is changed from R to eR, where e is a scaling factornot larger than 1, since the new set of Zernike coefficients for thedefocus term can be related to its original coefficients asb ₂ ⁰ =[c ₂ ⁰−√{square root over (15)}(1−e ²)c ₄ ⁰+√{square root over(21)}(2−5e ²+3e ⁴)c ₆ ⁰−√{square root over (3)}(10−45e ²+63e ⁴−28e ⁶)c ₈⁰ ]e ²  (8)Substituting c₂ ⁰ with b₂ ⁰, and R² with e²R² in Equation 1 usingEquation 8, we have4√{square root over (3)}c ₂ ⁰−12√{square root over (5)}(1−e ²)c ₄⁰+12√{square root over (7)}(2−5e ²+3e ⁴)c ₆ ⁰−12(10−45e ²+63e ⁴−28e ⁶)c₈ ⁰ =−R ² P  (9)Suppose we request power p₀ at radius e₀R, p₁ at radius e₁R, p₂ andradius e₂R, and p₃ and radius e₃R, an analytical solution of theoriginal wavefront shape, which is represented by c₂ ⁰, c₄ ⁰, c₆ ⁰ andc₈ ⁰, can be obtained as:

$\begin{matrix}{{{c_{2}^{0} = {{- R^{2}}\frac{{\alpha_{3}p_{0}} - {\beta_{3}p_{1}} + {\gamma_{3}p_{2}} - {\delta_{3}p_{3}}}{4\sqrt{3}\lambda}}}{c_{4}^{0} = {{- R^{2}}\frac{{\alpha_{2}p_{0}} - {\beta_{2}p_{1}} + {\gamma_{2}p_{2}} - {\delta_{2}p_{3}}}{252\sqrt{5}\lambda}}}{c_{6}^{0} = {{- R^{2}}\frac{{\alpha_{1}p_{0}} - {\beta_{1}p_{1}} + {\gamma_{1}p_{2}} - {\delta_{1}p_{3}}}{144\sqrt{7}\lambda}}}c_{8}^{0} = {{- R^{2}}\frac{{\alpha_{0}p_{0}} - {\beta_{0}p_{1}} + {\gamma_{0}p_{2}} - {\delta_{0}p_{3}}}{336\lambda}}},} & (10)\end{matrix}$where

$\begin{matrix}{\lambda = {\left( {e_{0}^{2} - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)\left( {e_{0}^{2} - e_{3}^{2}} \right)\left( {e_{1}^{2} - e_{2}^{2}} \right)\left( {e_{1}^{2} - e_{3}^{2}} \right)\left( {e_{2}^{2} - e_{3}^{2}} \right)}} & (11) \\{\alpha_{0} = {\left( {e_{1}^{2} - e_{2}^{2}} \right)\left( {e_{1}^{2} - e_{3}^{2}} \right)\left( {e_{2}^{2} - e_{3}^{2}} \right)}} & (12) \\{\beta_{0} = {\left( {e_{0}^{2} - e_{2}^{2}} \right)\left( {e_{0}^{2} - e_{3}^{2}} \right)\left( {e_{2}^{2} - e_{3}^{2}} \right)}} & (13) \\{\gamma_{0} = {\left( {e_{0}^{2} - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{3}^{2}} \right)\left( {e_{1}^{2} - e_{3}^{2}} \right)}} & (14) \\{\delta_{0} = {\left( {e_{0}^{2} - e_{1}^{2}} \right)\left( {e_{0}^{2} - e_{2}^{2}} \right)\left( {e_{1}^{2} - e_{2}^{2}} \right)}} & (15) \\{\alpha_{1} = {\left\lbrack {9 - {4\left( {e_{1}^{2} + e_{2}^{2} + e_{3}^{2}} \right)}} \right\rbrack\alpha_{0}}} & (16) \\{\beta_{1} = {\left\lbrack {9 - {4\left( {e_{0}^{2} + e_{2}^{2} + e_{3}^{2}} \right)}} \right\rbrack\beta_{0}}} & (17) \\{\gamma_{1} = {\left\lbrack {9 - {4\left( {e_{0}^{2} + e_{1}^{2} + e_{3}^{2}} \right)}} \right\rbrack\gamma_{0}}} & (18) \\{\delta_{1} = {\left\lbrack {9 - {4\left( {e_{0}^{2} + e_{1}^{2} + e_{2}^{2}} \right)}} \right\rbrack\delta_{0}}} & (19) \\{\alpha_{2} = {\left\lbrack {45 - {35\left( {e_{1}^{2} + e_{2}^{2} + e_{3}^{2}} \right)} + {21\left( {{e_{1}^{2}e_{2}^{2}} + {e_{1}^{2}e_{3}^{2}} + {e_{2}^{2}e_{3}^{2}}} \right)}} \right\rbrack\alpha_{0}}} & (20) \\{\beta_{2} = {\left\lbrack {45 - {35\left( {e_{0}^{2} + e_{2}^{2} + e_{3}^{2}} \right)} + {21\left( {{e_{0}^{2}e_{2}^{2}} + {e_{0}^{2}e_{3}^{2}} + {e_{2}^{2}e_{3}^{2}}} \right)}} \right\rbrack\beta_{0}}} & (21) \\{\gamma_{2} = {\left\lbrack {45 - {35\left( {e_{0}^{2} + e_{1}^{2} + e_{3}^{2}} \right)} + {21\left( {{e_{0}^{2}e_{1}^{2}} + {e_{0}^{2}e_{3}^{2}} + {e_{1}^{2}e_{3}^{2}}} \right)}} \right\rbrack\gamma_{0}}} & (22) \\{\delta_{2} = {\left\lbrack {45 - {35\left( {e_{0}^{2} + e_{1}^{2} + e_{2}^{2}} \right)} + {21\left( {{e_{0}^{2}e_{1}^{2}} + {e_{0}^{2}e_{2}^{2}} + {e_{1}^{2}e_{2}^{2}}} \right)}} \right\rbrack\delta_{0}}} & (23) \\{\alpha_{3} = {\left( {1 - e_{1}^{2}} \right)\left( {1 - e_{2}^{2}} \right)\left( {1 - e_{3}^{2}} \right)\alpha_{0}}} & (24) \\{\beta_{3} = {\left( {1 - e_{0}^{2}} \right)\left( {1 - e_{2}^{2}} \right)\left( {1 - e_{3}^{2}} \right)\beta_{0}}} & (25) \\{\gamma_{3} = {\left( {1 - e_{0}^{2}} \right)\left( {1 - e_{1}^{2}} \right)\left( {1 - e_{3}^{2}} \right)\gamma_{0}}} & (26) \\{\delta_{3} = {\left( {1 - e_{0}^{2}} \right)\left( {1 - e_{1}^{2}} \right)\left( {1 - e_{2}^{2}} \right)\delta_{0}}} & (27)\end{matrix}$

As an example, let's consider a pupil with WaveScan pupil size of 6 mm,and dim distance pupil size of 6 mm, requesting effective power of 0 Dand bright reading pupil size of 3.5 mm, requesting effective power of−1.5 D. We also request that the bright distance pupil size to be 5 mmand dim reading pupil size of 4.5 mm, with effective power of −0.2 D and−0.5 D, respectively. Substituting e₀=6/6=1, e₁=5/6=0.833, e₂=4.5/6=0.75and e₃=3.5/6=0.583 as well as p₀=0, p₁=−0.2, p₂=−0.5 and p₃=−1.5, we getc₂ ⁰=0, c₄ ⁰=−0.2919, c₆ ⁰=0.3523 and c₈ ⁰=−0.105. FIGS. 39 and 40 showthe presbyopia shape and the effective power as a function of pupilsizes. Note that both the presbyopia shape and the effective power aresimilar to those shown in FIGS. 37 and 38. However, the shape and powergiven with 4-term solution is smoother and have a flatter power atlarger pupil sizes.

It is also possible to use the same approach to obtain analyticalsolutions for conditions that use more than four power points. Forexample, when we use five power points, we could use up to 10^(th) orderof Zernike coefficients to describe the aspheric shape that satisfiesthe power profile defined with five power points. Similarly, six powerpoints can define an aspheric shape using 12^(th) order of Zernikecoefficients. Because more power points can in general make theanalytical solution more difficult, another way of approaching thesolution is by more complex numerical algorithms. Due to theavailability of the recursive formula, the equations that lead toanalytical solutions might be converted to an eigen system problem,which does have numerical solutions, optionally making use of themethods of William H Press, Saul A. Teukolsky, William Vetterling, andBrian P. Flannery, in Numerical Recipes in C++, (Cambridge UniversityPress, 2002). Such a solution may be more accurate than use of discretepower point.

Discussion

The first thing we want to discuss is how many terms we should use indetermining the presbyopia shape. In the two-power-term solution, we usethe pupil sizes as well as the corresponding desired powers. Obviously,we can use this solution for a somewhat “bifocal” design with onedistance pupil size and power (which should be zero to keep the eye atemmetropia) and one reading pupil size and its corresponding power. FromFIGS. 35 and 36, the effective power follows a rather linearrelationship with pupil size changes. This may not be ideal in that thedistance power may tend to become myopic. With a 3-power-term solution,we have one more freedom to choose the power in a middle pupil size andin fact the solution is rather close to a 4-power-term solution whencarefully designed. Unfortunately, with a 3-power-term solution, thebright distance pupil and the dim reading pupil tend to be averaged andso do the corresponding powers. This may become too inflexible to designan ideal shape. Therefore, the 4-power-term solution, which tends togive a more favorable reverse Z-curve, should be used in the practicalimplementation. The reverse Z-curve such as that shown in FIG. 41A, apositive power gradient region between two lower slope (or flat) regionswithin a pupil size variation range for a particular eye, may be abeneficial effective power characteristic for presbyopia mitigation.

Even with a 4-power-term solution, choosing effective powers in-betweendim distance pupil and bright reading pupil should be carefullyconsidered. For instance, in order to satisfy restaurant menu reading,we might want to increase the power for dim reading. In this case, anunfavorable S-curve would exist, as is also shown in FIG. 41A.Presbyopia-mitigation shapes corresponding to the S-curve and Z-curveshapes are shown in FIG. 41B. These results were generated for a 6 mmpupil with the dim distance pupil at 6 mm with a power of 0 D, thebright distance pupil at 5 mm with power of −0.2 D and −0.7 D, the dimreading pupil at 4.5 mm with a power of −1.2 D and the bright readingpupil at 3.5 mm with a power of −1.5 D. To reduce the fluctuation ofeffective power, we can also increase the power in bright distance andin this case the distance vision can be affected (in addition to thecontrast drop due to asphericity).

Another parameter we can set is desired reading power. Optionally we cangive the patient full power; say 2.5 D, so the treatment can besufficient to treat presbyopia for the patient's life span. However, thenatural pupil size decreases with increasing age. Therefore, a shapewell suited to a patient at the age of 45 could become deleterious atthe age of 60. Secondly, not everyone easily tolerates asphericity.Furthermore, too much asphericity can reduce the contrast sensitivity toa level that distance vision would deteriorate. As such, measurement ofa patient's residual accommodation becomes beneficial in the success ofpresbyopia correction. In addition, the various pupil sizes at differentlighting conditions and accommodation can be measured systematically andmore accurately. Such measurements may employ, for example, acommercially available pupilometer sold by PROCYON under the modelnumber P-2000. A wide variety of alternative pupil measurementtechniques might be used, including visual measurements, optionallyusing a microscope displaying a scale and/or reticule of known sizesuperimposed on the eye, similar to those employed on laser eye surgerysystems commercially available from VISX, INCORPORATED of Santa Clara,Calif.

The influence of high order aberrations on the effective power, asdescribed above regarding the power map, may also be incorporated intothe presbyopia-mitigating shape calculations. This may involveintegration over the entire power map, i.e., the average power, withappropriate adjustment so as to avoid overestimating power (that mayotherwise not agree with the minimum root-mean-squares (RMS) criterion)and so as to correlate with patient data. The influence of high orderspherical aberrations on effective power calculation should not beentirely ignored. In particular, the influence on the depth of focus,and hence to the blur range during manifest refraction test, can bedetermined using clinical testing.

Taking advantage of the ability to calculate presbyopia shapes based oneffective power, presbyopia-mitigating shapes can be derived and/oroptimized based on the following considerations. First, image quality ofthe presbyopia shape at different viewing conditions can be evaluated.To do so, optimization of the shape itself can be pursued. This can bedone in several ways, such as using diffraction optics (wave optics) orgeometrical optics (ray tracing). Because we are dealing withaberrations of many waves, it may be impractical to use point spreadfunction based optical metrics. However, since the aberration weintroduce belongs to high orders only, wave optics may still work well.In fact, a comparison of Zemax modeling with three wavelengths and usingverification tools (wave optics), as shown in FIG. 13, with7-wavelengths show almost identical results in both point spreadfunction (PSF) and modulation transfer function (MTF). FIG. 42 showssome derived shapes for a 5 mm and a 6 mm pupil, while the correspondingMTF curves are shown in FIG. 43. The simulated blurring of eye chartletter E for both cases is shown in FIG. 44. These letters graphicallyillustrate verification of presbyopia shape using a goal function with7-wavelengths polychromatic PSF and a 20/20 target. The first imageshows a target at 10 m. The second to the last image shows targets from1 m to 40 cm, separated by 0.1 D in vergence. One diopter of residualaccommodation is assumed for each. Even without optimization, theoptical surface shown gives almost 20/20 visual acuity over 1.5 Dvergence.

The above approach is valid to apply in contact lens, intra-ocular lens,as well as spectacles, as well as refractive surgery. Such calculationsfor refractive surgery may be adjusted for the healing effect as well asthe LASIK flap effect based on empirical studies and clinicalexperience.

As established above, it is possible to obtain analytical expressionsfor the Zernike coefficients of the first few spherical aberrations ofdifferent orders to create an aspheric shape for presbyopia correctionbased on one or more desired effective powers. Healing effect, flapeffect, and the correlation of effective power with manifest refractionwill benefit from additional patient data and empirical studies tofurther refine the presbyopia shape so as to (for example) moreaccurately plan the shape for future ablation.

Each of the above calculations may be performed using a computer orother processor having hardware, software, and/or firmware. The variousmethod steps may be performed by modules, and the modules may compriseany of a wide variety of digital and/or analog data processing hardwareand/or software arranged to perform the method steps described herein.The modules optionally comprising data processing hardware adapted toperform one or more of these steps by having appropriate machineprogramming code associated therewith, the modules for two or more steps(or portions of two or more steps) being integrated into a singleprocessor board or separated into different processor boards in any of awide variety of integrated and/or distributed processing architectures.These methods and systems will often employ a tangible media embodyingmachine-readable code with instructions for performing the method stepsdescribed above. Suitable tangible media may comprise a memory(including a volatile memory and/or a non-volatile memory), a storagemedia (such as a magnetic recording on a floppy disk, a hard disk, atape, or the like; on an optical memory such as a CD, a CD-R/W, aCD-ROM, a DVD, or the like; or any other digital or analog storagemedia), or the like.

As the analytical solutions described herein some or all of these methodsteps may be performed with computer processors of modest capability,i.e., a 386 processor from Intel™ may be enough to calculate the Zernikecoefficients, and even 286 processor may be fine. Scaling of Zernikecoefficients was described by Jim Schweigerling, “Scaling ZernikeExpansion Coefficients to Different Pupil Sizes,” J. Opt. Soc. Am. A 19,pp 1937-1945 (2002). No special memory is needed (i.e., no buffers, allcan be done as regular variables or using registers). Also, it can bewritten in any of a wide variety of computer languages, with theexemplary embodiment employing C++. This exemplary embodiment comprisescode which performs the Zernike coefficient calculation, shapecombination (combining a regular aberration treatment prescription aswell as the presbyopia shape), and provides graphical output forreporting purpose. It was written in C++ with Borland C++ Builder™ 6,and it is run with a laptop of 1.13 GHz CPU having 512 Mb of memory.

FIGS. 45A and 45B illustrate exemplary desired power curves andtreatment shapes for mitigating presbyopia of a particular patient. Thefour power point solution was used to establish these shapes. For a 6 mmpupil, the following table describes the four conditions or set pointsfrom which the shape was generated:

TABLE 5 6 mm Pupil 5 mm Pupil Effective Pupil size Effective Pupil sizeConditions power (mm) power (mm) 1 0 D 6 0 D 5 2 −0.5 D 5 −0.5 D 4.2 3−1 D 4.5 −1 D 3.8 4 −1.5 D 3.4 −1.55 D 3.1

FIG. 45A shows the effective power profiles, while FIG. 45B shows thecorresponding presbyopia shapes. To model the healing and LASIK flapeffect, we uniformly boost the shape by 15%. In addition to the addedpresbyopia shape, we also used −0.6 D physician adjustment in thewavefront prescription generation to offset myopic bias to aimemmetropia at normal viewing condition (bright distance) after surgery.

While the exemplary embodiments have been described in some detail, byway of example and for clarity of understanding, those of skill in theart will recognize that a variety of modification, adaptations, andchanges may be employed. Hence, the scope of the present inventionshould be limited solely by the claims.

1. A method for identifying an intra-ocular lens that mitigates ortreats presbyopia of an eye in a particular patient, the methodcomprising: inputting a set of patient parameters specific for theparticular patient; determining a prescription for the eye based on theset of patient parameters per a goal function so as to mitigate or treatthe presbyopia in the patient; and identifying an intra-ocular lensshaped to impose the prescription on the eye when the intra-ocular lensis implanted into the eye.
 2. The method of claim 1, wherein the goalfunction reflects optical quality throughout a vergence range.
 3. Themethod of claim 1, wherein the goal function comprises a ratio of anoptical parameter of the eye with a diffraction theory parameter.
 4. Themethod of claim 3, wherein the goal function comprises at least oneparameter selected from the group consisting of Strehl Ratio (SR),modulation transfer function (MTF), point spread function (PSF),encircled energy (EE), MTF volume or volume under MTF surface (MTFV),compound modulation transfer function (CMTF), and contrast sensitivity(CS).
 5. The method of claim 4, wherein the prescription is determinedsuch that it has a value of about 25% CMTF over a vergence.
 6. Themethod of claim 4, further comprising creating the intra-ocular lens. 7.The method of claim 1, further comprising creating the intra-ocularlens.
 8. The method of claim 1, wherein the goal function is based ongeometrical optics.
 9. The method of claim 8, wherein the goal functionis determined using ray tracing.
 10. The method of claim 1, wherein theset of patient parameters comprises at least one parameter selected fromthe group consisting of pupil size, residual accommodation, and desiredpower.
 11. The method of claim 10, further wherein additional patientparameters comprise at least one of the group consisting of preferencesfor distance or near sight, preferences for sight under bright or dimconditions, and preferences for contrast sensitivity.
 12. The method ofclaim 1, wherein the prescription is further determined based on anexpansion selected from the group consisting of a regular polynomial(EPTP or non-EPTP), a Zernike polynomial, a Fourier series, and adiscrete shape entirety.
 13. The method of claim 12, wherein theexpansion is a 3rd order or 4th order non-EPTP expansion.
 14. The methodof claim 12, wherein the expansion is a 6th order or 8th order EPTPexpansion.
 15. The method of claim 1, wherein the prescription isfurther determined based on a presbyopia-add to pupil ratio (PAR), thePAR ranging from about 0.2 to about 1.0.
 16. The method of claim 1,wherein the prescription has an optimizer value of about 5.0 or smaller.17. A system for identifying an intra-ocular lens that mitigates ortreats presbyopia of an eye in a particular patient, the systemcomprising: an input that accepts a set of patient parameters; and amodule comprising a tangible medium embodying machine-readable code thatidentifies the intra-ocular lens, the intra-ocular lens shaped to imposea prescription on the eye when the intra-ocular lens is implanted intothe eye, the prescription based on the set of patient parameters per agoal function so as to mitigate or treat the presbyopia in the patient.18. The system of claim 17, wherein the goal function reflects opticalquality throughout a vergence range.
 19. The system of claim 17, whereinthe goal function comprises a ratio of an optical parameter of the eyewith a diffraction theory parameter.
 20. The system of claim 19, whereinthe goal function comprises at least one parameter selected from thegroup consisting of Strehl Ratio (CS), modulation transfer function(MTF), point spread function (PSF), encircled energy (EE), MTF volume orvolume under MTF surface (MTFV), compound modulation transfer function(CMTF), and contrast sensitivity (CS).
 21. The system of claim 20,further comprising an intra-ocular lens fabrication system.
 22. Thesystem of claim 17, further comprising an intra-ocular lens fabricationsystem.
 23. The system of claim 17, wherein the goal function is basedon geometrical optics.
 24. The system of claim 23, wherein the goalfunction is determined using ray tracing.
 25. The system of claim 17,wherein the set of patient parameters comprises at least one parameterselected from the group consisting of pupil size, residualaccommodation, and power need.
 26. The system of claim 17, wherein themodule comprises a tangible medium embodying machine-readable code thatmakes use of an initial optical shape, a set of initial conditions, andthe set of patient parameters for an iterative optimization so as toestablish an optical shape for the intra-ocular lens, using a goalfunction appropriate for presbyopia of an eye.
 27. The system of claim26, further comprising an intra-ocular lens fabrication system.
 28. Thesystem of claim 26, wherein the initial optical shape is radiallysymmetric.
 29. The system of claim 28, wherein the radially symmetricshape is decomposed by the module into a set of polynomials having atleast two independent variables.
 30. The system of claim 29, wherein oneof the at least two independent variables is the ratio of the customizedshape diameter to pupil diameter.
 31. The system of claim 26, whereinthe optimization code is configured to employ a method selected from thegroup consisting of Downhill Simplex optimization, Direction setoptimization, and Simulated Annealing optimization.
 32. The system ofclaim 26, wherein the set of patient parameters comprises at least oneparameter selected from the group consisting of pupil size, residualaccommodation, and desired power.